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id="toc-Notation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notation"> <div class="vector-toc-text"> 2.1 Notation </div> </a> <ul id="toc-Notation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Construction_of_new_semirings" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Construction_of_new_semirings"> <div class="vector-toc-text"> 3 Construction of new semirings </div> </a> <button aria-controls="toc-Construction_of_new_semirings-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Construction of new semirings subsection </button> <ul id="toc-Construction_of_new_semirings-sublist" class="vector-toc-list"> <li id="toc-Derivations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Derivations"> <div class="vector-toc-text"> 3.1 Derivations </div> </a> <ul id="toc-Derivations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> 4 Properties </div> </a> <button aria-controls="toc-Properties-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Properties subsection </button> <ul id="toc-Properties-sublist" class="vector-toc-list"> <li id="toc-Semifields" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Semifields"> <div class="vector-toc-text"> 4.1 Semifields </div> </a> <ul id="toc-Semifields-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rings" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rings"> <div class="vector-toc-text"> 4.2 Rings </div> </a> <ul id="toc-Rings-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Commutative_semirings" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Commutative_semirings"> <div class="vector-toc-text"> 4.3 Commutative semirings </div> </a> <ul id="toc-Commutative_semirings-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ordered_semirings" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ordered_semirings"> <div class="vector-toc-text"> 4.4 Ordered semirings </div> </a> <ul id="toc-Ordered_semirings-sublist" class="vector-toc-list"> <li id="toc-Additively_idempotent_semirings" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Additively_idempotent_semirings"> <div class="vector-toc-text"> 4.4.1 Additively idempotent semirings </div> </a> <ul id="toc-Additively_idempotent_semirings-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Number_lines" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Number_lines"> <div class="vector-toc-text"> 4.4.2 Number lines </div> </a> <ul id="toc-Number_lines-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Discretely_ordered_semirings" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Discretely_ordered_semirings"> <div class="vector-toc-text"> 4.4.3 Discretely ordered semirings </div> </a> <ul id="toc-Discretely_ordered_semirings-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Natural_numbers" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Natural_numbers"> <div class="vector-toc-text"> 4.4.4 Natural numbers </div> </a> <ul id="toc-Natural_numbers-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Complete_semirings" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Complete_semirings"> <div class="vector-toc-text"> 4.5 Complete semirings </div> </a> <ul id="toc-Complete_semirings-sublist" class="vector-toc-list"> <li id="toc-Continuous_semirings" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Continuous_semirings"> <div class="vector-toc-text"> 4.5.1 Continuous semirings </div> </a> <ul id="toc-Continuous_semirings-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Star_semirings" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Star_semirings"> <div class="vector-toc-text"> 4.6 Star semirings </div> </a> <ul id="toc-Star_semirings-sublist" class="vector-toc-list"> <li id="toc-Complete_star_semirings" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Complete_star_semirings"> <div class="vector-toc-text"> 4.6.1 Complete star semirings </div> </a> <ul id="toc-Complete_star_semirings-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Conway_semiring" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Conway_semiring"> <div class="vector-toc-text"> 4.6.2 Conway semiring </div> </a> <ul id="toc-Conway_semiring-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> 5 Examples </div> </a> <button aria-controls="toc-Examples-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Examples subsection </button> <ul id="toc-Examples-sublist" class="vector-toc-list"> <li id="toc-Star_semirings_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Star_semirings_2"> <div class="vector-toc-text"> 5.1 Star semirings </div> </a> <ul id="toc-Star_semirings_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> 6 Applications </div> </a> <ul id="toc-Applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalizations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Generalizations"> <div class="vector-toc-text"> 7 Generalizations </div> </a> <ul id="toc-Generalizations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 8 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 9 Notes </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Citations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Citations"> <div class="vector-toc-text"> 10 Citations </div> </a> <ul id="toc-Citations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliography" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Bibliography"> <div class="vector-toc-text"> 11 Bibliography </div> </a> <ul id="toc-Bibliography-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> 12 Further reading </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" 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For other uses, see <a href="/wiki/Ring_of_sets#semiring" title="Ring of sets">Ring of sets § semiring</a>.</div> <div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Algebraic ring that need not have additive negative elements</div> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist 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.sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist" style="width: 20.5em;"><tbody><tr><th class="sidebar-title" style="padding-bottom:0.4em;"><a href="/wiki/Algebraic_structure" title="Algebraic structure">Algebraic structure</a> → Ring theory <a href="/wiki/Ring_theory" title="Ring theory">Ring theory</a></th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Basic concepts</div><div class="sidebar-list-content mw-collapsible-content" style="text-align: left;"><a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">Rings</a> <dl><dd>• <a href="/wiki/Subring" title="Subring">Subrings</a></dd> <dd>• <a href="/wiki/Ideal_(ring_theory)" title="Ideal (ring theory)">Ideal</a></dd> <dd>• <a href="/wiki/Quotient_ring" title="Quotient ring">Quotient ring</a> <dl><dd>• <a href="/wiki/Fractional_ideal" title="Fractional ideal">Fractional ideal</a></dd> <dd>• <a href="/wiki/Total_ring_of_fractions" title="Total ring of fractions">Total ring of fractions</a></dd></dl></dd> <dd>• <a href="/wiki/Product_of_rings" title="Product of rings">Product of rings</a></dd> <dd>• <a href="/wiki/Free_product_of_associative_algebras" title="Free product of associative algebras">Free product of associative algebras</a></dd> <dd>• <a href="/wiki/Tensor_product_of_algebras" title="Tensor product of algebras">Tensor product of algebras</a></dd></dl> <a href="/wiki/Ring_homomorphism" title="Ring homomorphism">Ring homomorphisms</a> <dl><dd>• <a href="/wiki/Kernel_(algebra)#Ring_homomorphisms" title="Kernel (algebra)">Kernel</a></dd> <dd>• <a href="/wiki/Inner_automorphism#Ring_case" title="Inner automorphism">Inner automorphism</a></dd> <dd>• <a href="/wiki/Frobenius_endomorphism" title="Frobenius endomorphism">Frobenius endomorphism</a></dd></dl> <a href="/wiki/Algebraic_structure" title="Algebraic structure">Algebraic structures</a> <dl><dd>• <a href="/wiki/Module_(mathematics)" title="Module (mathematics)">Module</a></dd> <dd>• <a href="/wiki/Associative_algebra" title="Associative algebra">Associative algebra</a></dd> <dd>• <a href="/wiki/Graded_ring" title="Graded ring">Graded ring</a></dd> <dd>• <a href="/wiki/Involutive_ring" class="mw-redirect" title="Involutive ring">Involutive ring</a></dd> <dd>• <a href="/wiki/Category_of_rings" title="Category of rings">Category of rings</a> <dl><dd>• <a href="/wiki/Integer" title="Integer">Initial ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" /></dd> <dd>• <a href="/wiki/Zero_ring" title="Zero ring">Terminal ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0=\mathbb {Z} /1\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0=\mathbb {Z} /1\mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e45ab495cb8cfbac68a9322af662c3d6c7dbe494" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.686ex; height:2.843ex;" alt="{\displaystyle 0=\mathbb {Z} /1\mathbb {Z} }" /></dd></dl></dd></dl> Related structures <dl><dd>• <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">Field</a> <dl><dd>• <a href="/wiki/Finite_field" title="Finite field">Finite field</a></dd></dl></dd> <dd>• <a href="/wiki/Non-associative_ring" class="mw-redirect" title="Non-associative ring">Non-associative ring</a> <dl><dd>• <a href="/wiki/Lie_ring" class="mw-redirect" title="Lie ring">Lie ring</a></dd> <dd>• <a href="/wiki/Jordan_ring" class="mw-redirect" title="Jordan ring">Jordan ring</a></dd></dl></dd> <dd>• <a class="mw-selflink selflink">Semiring</a> <dl><dd>• <a href="/wiki/Semifield" title="Semifield">Semifield</a></dd></dl></dd></dl></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Commutative_algebra" title="Commutative algebra">Commutative algebra</a></div><div class="sidebar-list-content mw-collapsible-content" style="text-align: left;"><a href="/wiki/Commutative_ring" title="Commutative ring">Commutative rings</a> <dl><dd>• <a href="/wiki/Integral_domain" title="Integral domain">Integral domain</a> <dl><dd>• <a href="/wiki/Integrally_closed_domain" title="Integrally closed domain">Integrally closed domain</a></dd> <dd>• <a href="/wiki/GCD_domain" title="GCD domain">GCD domain</a></dd> <dd>• <a href="/wiki/Unique_factorization_domain" title="Unique factorization domain">Unique factorization domain</a></dd> <dd>• <a href="/wiki/Principal_ideal_domain" title="Principal ideal domain">Principal ideal domain</a></dd> <dd>• <a href="/wiki/Euclidean_domain" title="Euclidean domain">Euclidean domain</a></dd> <dd>• <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">Field</a> <dl><dd>• <a href="/wiki/Finite_field" title="Finite field">Finite field</a></dd></dl></dd> <dd>• <a href="/wiki/Polynomial_ring" title="Polynomial ring">Polynomial ring</a></dd> <dd>• <a href="/wiki/Formal_power_series_ring" class="mw-redirect" title="Formal power series ring">Formal power series ring</a></dd></dl></dd></dl> <a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">Algebraic number theory</a> <dl><dd>• <a href="/wiki/Algebraic_number_field" title="Algebraic number field">Algebraic number field</a></dd> <dd>• <a href="/wiki/Integers_modulo_n" class="mw-redirect" title="Integers modulo n">Integers modulo n</a></dd> <dd>• <a href="/wiki/Ring_of_integers" title="Ring of integers">Ring of integers</a></dd> <dd>• <a href="/wiki/P-adic_integer" class="mw-redirect" title="P-adic integer">p-adic integers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbc1df7227ef11fe88dccd2dae3adc7bbdeae5f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.609ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} _{p}}" /></dd> <dd>• <a href="/wiki/P-adic_number" title="P-adic number">p-adic numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f44bc6894c682710705f3ea74f33042e0acc3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.867ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} _{p}}" /></dd> <dd>• <a href="/wiki/Pr%C3%BCfer_group#The_Prüfer_group_as_a_ring" title="Prüfer group">Prüfer p-ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} (p^{\infty })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">(</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} (p^{\infty })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14af623e08c241266c125ad927dd35086ec8ce90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.404ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} (p^{\infty })}" /></dd></dl></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Noncommutative_algebra" class="mw-redirect" title="Noncommutative algebra">Noncommutative algebra</a></div><div class="sidebar-list-content mw-collapsible-content" style="text-align: left;"><a href="/wiki/Noncommutative_ring" title="Noncommutative ring">Noncommutative rings</a> <dl><dd>• <a href="/wiki/Division_ring" title="Division ring">Division ring</a></dd> <dd>• <a href="/wiki/Semiprimitive_ring" title="Semiprimitive ring">Semiprimitive ring</a></dd> <dd>• <a href="/wiki/Simple_ring" title="Simple ring">Simple ring</a></dd> <dd>• <a href="/wiki/Commutator_(ring_theory)" class="mw-redirect" title="Commutator (ring theory)">Commutator</a></dd></dl> <a href="/wiki/Noncommutative_algebraic_geometry" title="Noncommutative algebraic geometry">Noncommutative algebraic geometry</a> <a href="/wiki/Free_algebra" title="Free algebra">Free algebra</a> <a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebra</a> <dl><dd>• <a href="/wiki/Geometric_algebra" title="Geometric algebra">Geometric algebra</a></dd></dl> <a href="/wiki/Operator_algebra" title="Operator algebra">Operator algebra</a></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Ring_theory_sidebar" title="Template:Ring theory sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Ring_theory_sidebar" title="Template talk:Ring theory sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Ring_theory_sidebar" title="Special:EditPage/Template:Ring theory sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> In <a href="/wiki/Abstract_algebra" title="Abstract algebra">abstract algebra</a>, a semiring is an <a href="/wiki/Algebraic_structure" title="Algebraic structure">algebraic structure</a>. Semirings are a generalization of <a href="/wiki/Ring_(algebra)" class="mw-redirect" title="Ring (algebra)">rings</a>, dropping the requirement that each element must have an <a href="/wiki/Additive_inverse" title="Additive inverse">additive inverse</a>. At the same time, semirings are a generalization of <a href="/wiki/Lattice_(order)#Bounded_lattice" title="Lattice (order)">bounded</a> <a href="/wiki/Distributive_lattice" title="Distributive lattice">distributive lattices</a>. The smallest semiring that is not a ring is the <a href="/wiki/Two-element_Boolean_algebra" title="Two-element Boolean algebra">two-element Boolean algebra</a>, for instance with <a href="/wiki/Logical_disjunction" title="Logical disjunction">logical disjunction</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lor }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab47f6b1f589aedcf14638df1d63049d233d851a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \lor }" /> as addition. A motivating example that is neither a ring nor a lattice is the set of <a href="/wiki/Natural_number" title="Natural number">natural numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }" /> (including zero) under ordinary addition and multiplication. Semirings are abundant because a suitable multiplication operation arises as the <a href="/wiki/Function_composition" title="Function composition">function composition</a> of <a href="/wiki/Endomorphism" title="Endomorphism">endomorphisms</a> over any <a href="/wiki/Commutative_monoid" class="mw-redirect" title="Commutative monoid">commutative monoid</a>. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" 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property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Terminology">Terminology</h2>[<a href="/w/index.php?title=Semiring&action=edit&section=1" title="Edit section: Terminology">edit</a>]</div> Some authors define semirings without the requirement for there to be a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" /> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" />. This makes the analogy between ring and semiring on the one hand and <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> and <a href="/wiki/Semigroup" title="Semigroup">semigroup</a> on the other hand work more smoothly. These authors often use rig for the concept defined here.<a href="#cite_note-FOOTNOTEGłazek20027-1">[1]</a><a href="#cite_note-2">[a]</a> This originated as a joke, suggesting that rigs are rings without negative elements. (Akin to using <a href="/wiki/Rng_(algebra)" title="Rng (algebra)">rng</a> to mean a ring without a multiplicative identity.) The term dioid (for "double monoid") has been used to mean semirings or other structures. It was used by Kuntzmann in 1972 to denote a semiring.<a href="#cite_note-3">[2]</a> (It is alternatively sometimes used for <a href="/wiki/Naturally_ordered_semiring" class="mw-redirect" title="Naturally ordered semiring">naturally ordered semirings</a><a href="#cite_note-4">[3]</a> but the term was also used for idempotent subgroups by <a href="/wiki/Fran%C3%A7ois_Baccelli" title="François Baccelli">Baccelli</a> et al. in 1992.<a href="#cite_note-5">[4]</a>) <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2>[<a href="/w/index.php?title=Semiring&action=edit&section=2" title="Edit section: Definition">edit</a>]</div> A semiring is a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" /> equipped with two <a href="/wiki/Binary_operation" title="Binary operation">binary operations</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cdot ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋅</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cdot ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ad72844e0986e4c8e59488a79a3a163a4385384" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.294ex; height:1.676ex;" alt="{\displaystyle \cdot ,}" /> called addition and multiplication, such that:<a href="#cite_note-FOOTNOTEBerstelPerrin1985[httpsbooksgooglecombooksidGHJHqezwwpcCpgPA26dq22asemiringKisasetequippedwithtwooperations22_p._26]-6">[5]</a><a href="#cite_note-FOOTNOTELothaire2005211-7">[6]</a><a href="#cite_note-FOOTNOTESakarovitch200927–28-8">[7]</a> <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (R,+)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>R</mi> <mo>,</mo> <mo>+</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (R,+)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/466356a631ac93bc70fbe2d276117d22f980e285" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.415ex; height:2.843ex;" alt="{\displaystyle (R,+)}" /> is a <a href="/wiki/Commutative" class="mw-redirect" title="Commutative">commutative</a> <a href="/wiki/Monoid" title="Monoid">monoid</a> with an <a href="/wiki/Identity_element" title="Identity element">identity element</a> called <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" />: <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a+b)+c=a+(b+c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>c</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a+b)+c=a+(b+c)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46b7b8d31d5845966e6abdbb030c73f343c17d4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.547ex; height:2.843ex;" alt="{\displaystyle (a+b)+c=a+(b+c)}" /></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0+a=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>+</mo> <mi>a</mi> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0+a=a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba565116cf2f2d984f7b8365b054b70eb8f89308" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.561ex; height:2.343ex;" alt="{\displaystyle 0+a=a}" /></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+0=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mn>0</mn> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+0=a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4564e28f0f8274644ca4e58664c0593ed48de541" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.561ex; height:2.343ex;" alt="{\displaystyle a+0=a}" /></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b=b+a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>=</mo> <mi>b</mi> <mo>+</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b=b+a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/684f43b5094501674e8314be5e24a80ee64682e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.234ex; height:2.343ex;" alt="{\displaystyle a+b=b+a}" /></li></ul></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (R,\,\cdot \,)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>R</mi> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mo>⋅</mo> <mspace width="thinmathspace"></mspace> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (R,\,\cdot \,)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e617cc3aa910126c4e10421a44229360667650c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.028ex; height:2.843ex;" alt="{\displaystyle (R,\,\cdot \,)}" /> is a monoid with an identity element called <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" />: <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a\cdot b)\cdot c=a\cdot (b\cdot c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>c</mi> <mo>=</mo> <mi>a</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>⋅</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a\cdot b)\cdot c=a\cdot (b\cdot c)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e48cd711b9a1eb1c3a2372ba01fa48ca7e262a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.902ex; height:2.843ex;" alt="{\displaystyle (a\cdot b)\cdot c=a\cdot (b\cdot c)}" /></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\cdot a=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>⋅</mo> <mi>a</mi> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\cdot a=a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29c15b773211eeb31e5a62eabd1a03f4b4719f56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.4ex; height:2.176ex;" alt="{\displaystyle 1\cdot a=a}" /></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot 1=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mn>1</mn> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot 1=a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3e5c8aeb598f9dadf4767a03328e05849f37035" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.4ex; height:2.176ex;" alt="{\displaystyle a\cdot 1=a}" /></li></ul></li></ul> Further, the following axioms tie to both operations: <ul><li>Through multiplication, any element is left- and right-<a href="/wiki/Annihilating_element" class="mw-redirect" title="Annihilating element">annihilated</a> by the additive identity: <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\cdot a=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>⋅</mo> <mi>a</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\cdot a=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfd10467dcb093cd67ab74eb5262e775d9b0d291" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.332ex; height:2.176ex;" alt="{\displaystyle 0\cdot a=0}" /></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot 0=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mn>0</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot 0=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31f39efe37450f6bce024486f458a90d11a08278" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.332ex; height:2.176ex;" alt="{\displaystyle a\cdot 0=0}" /></li></ul></li> <li>Multiplication left- and right-<a href="/wiki/Distributive_law" class="mw-redirect" title="Distributive law">distributes</a> over addition: <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot (b+c)=(a\cdot b)+(a\cdot c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>⋅</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot (b+c)=(a\cdot b)+(a\cdot c)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0969d65db9f1f1097aa4f72bcddac8c46f1ca6ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.943ex; height:2.843ex;" alt="{\displaystyle a\cdot (b+c)=(a\cdot b)+(a\cdot c)}" /></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (b+c)\cdot a=(b\cdot a)+(c\cdot a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>a</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>⋅</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>⋅</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (b+c)\cdot a=(b\cdot a)+(c\cdot a)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56a9c8142f94470bb371538dfd87cae80f01b286" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.943ex; height:2.843ex;" alt="{\displaystyle (b+c)\cdot a=(b\cdot a)+(c\cdot a)}" /></li></ul></li></ul> <div class="mw-heading mw-heading3"><h3 id="Notation">Notation</h3>[<a href="/w/index.php?title=Semiring&action=edit&section=3" title="Edit section: Notation">edit</a>]</div> The symbol <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cdot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋅</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cdot }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba2c023bad1bd39ed49080f729cbf26bc448c9ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.439ex; margin-bottom: -0.61ex; width:0.647ex; height:1.176ex;" alt="{\displaystyle \cdot }" /> is usually omitted from the notation; that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/620419d3ed53abc98659a5fc0f3a5eb6177830ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.906ex; height:2.176ex;" alt="{\displaystyle a\cdot b}" /> is just written <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ab.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>b</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ab.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b39b72e3b30fae0a7f9b6086013251f3d87f523d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.874ex; height:2.176ex;" alt="{\displaystyle ab.}" /> Similarly, an <a href="/wiki/Order_of_operations" title="Order of operations">order of operations</a> is conventional, in which <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cdot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋅</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cdot }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba2c023bad1bd39ed49080f729cbf26bc448c9ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.439ex; margin-bottom: -0.61ex; width:0.647ex; height:1.176ex;" alt="{\displaystyle \cdot }" /> is applied before <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}" />. That is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b\cdot c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>⋅</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b\cdot c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53b41e6ad082d46b600768c8890598479b651d13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.754ex; height:2.343ex;" alt="{\displaystyle a+b\cdot c}" /> denotes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+(b\cdot c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>⋅</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+(b\cdot c)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3721810f5f940defbe23560ac5792efc1abc0963" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.563ex; height:2.843ex;" alt="{\displaystyle a+(b\cdot c)}" />. For the purpose of disambiguation, one may write <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0_{R}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0_{R}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/911229219f2ec31e016369e0ea56eef4b1f6b0c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.642ex; height:2.509ex;" alt="{\displaystyle 0_{R}}" /> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1_{R}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1_{R}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4b84310fba9b328ce04e95ee5aeaf1c81236785" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.642ex; height:2.509ex;" alt="{\displaystyle 1_{R}}" /> to emphasize which structure the units at hand belong to. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b4a3f5fa1b895f5a40a25ced8581b2152b3c24c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.934ex; height:2.176ex;" alt="{\displaystyle x\in R}" /> is an element of a semiring and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in {\mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in {\mathbb {N} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5d9ab68f55d202b14c6d41969d53ed7e245c47d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in {\mathbb {N} }}" />, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" />-times repeated multiplication of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" /> with itself is denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/150d38e238991bc4d0689ffc9d2a852547d2658d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.548ex; height:2.343ex;" alt="{\displaystyle x^{n}}" />, and one similarly writes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\,n:=x+x+\cdots +x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mi>n</mi> <mo>:=</mo> <mi>x</mi> <mo>+</mo> <mi>x</mi> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\,n:=x+x+\cdots +x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f75b63f56a767e417b570ba9fa51554de7fe56f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:22.09ex; height:2.176ex;" alt="{\displaystyle x\,n:=x+x+\cdots +x}" /> for the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" />-times repeated addition. <div class="mw-heading mw-heading2"><h2 id="Construction_of_new_semirings">Construction of new semirings</h2>[<a href="/w/index.php?title=Semiring&action=edit&section=4" title="Edit section: Construction of new semirings">edit</a>]</div> The <a href="/wiki/Zero_ring" title="Zero ring">zero ring</a> with underlying set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{0\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ff0df9ef65c0572eb676580ce1c02b8ec40f694" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle \{0\}}" /> is a semiring called the trivial semiring. This triviality can be characterized via <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2bb1a7bcdbd7e274b7b8581e4357f09dbeee7fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.423ex; height:2.176ex;" alt="{\displaystyle 0=1}" /> and so when speaking of nontrivial semirings, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\neq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≠</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\neq 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5add705e86314a6ce57c76d7493896b092661a75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.423ex; height:2.676ex;" alt="{\displaystyle 0\neq 1}" /> is often silently assumed as if it were an additional axiom. Now given any semiring, there are several ways to define new ones. As noted, the natural numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbb {N} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bbdf4a20a0a0a50daaa524863051d5d22c9fb23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle {\mathbb {N} }}" /> with its arithmetic structure form a semiring. Taking the zero and the image of the successor operation in a semiring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" />, i.e., the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x\in R\mid x=0_{R}\lor \exists p.x=p+1_{R}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mi>R</mi> <mo>∣</mo> <mi>x</mi> <mo>=</mo> <msub> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo>∨</mo> <mi mathvariant="normal">∃</mi> <mi>p</mi> <mo>.</mo> <mi>x</mi> <mo>=</mo> <mi>p</mi> <mo>+</mo> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{x\in R\mid x=0_{R}\lor \exists p.x=p+1_{R}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e03a6e3bb6ec49f23c5b8f47dce66a93c453e67a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.425ex; height:2.843ex;" alt="{\displaystyle \{x\in R\mid x=0_{R}\lor \exists p.x=p+1_{R}\}}" /> together with the inherited operations, is always a sub-semiring of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" />. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (M,+)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>M</mi> <mo>,</mo> <mo>+</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (M,+)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac85e31c15b188cc6674c500245dd710fc78d8ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.094ex; height:2.843ex;" alt="{\displaystyle (M,+)}" /> is a commutative monoid, function composition provides the multiplication to form a semiring: The set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {End} (M)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>End</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {End} (M)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a654fdfacc063d9cb412bfac7b28eb2198b6f909" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.419ex; height:2.843ex;" alt="{\displaystyle \operatorname {End} (M)}" /> of endomorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\to M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo stretchy="false">→</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M\to M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/864328478ca8418b4a695a59f7e66f7428d375e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.498ex; height:2.176ex;" alt="{\displaystyle M\to M}" /> forms a semiring where addition is defined from pointwise addition in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" />. The <a href="/wiki/Zero_morphism" title="Zero morphism">zero morphism</a> and the identity are the respective neutral elements. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M=R^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M=R^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63405a70e1240cfc0ca9e97b9431d1d65d22869b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.523ex; height:2.343ex;" alt="{\displaystyle M=R^{n}}" /> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" /> a semiring, we obtain a semiring that can be associated with the square <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}" /> <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}}_{n}(R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}_{n}(R)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/898fd3711a7daf6e6998854deaf83fe985ffd896" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.582ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M}}_{n}(R)}" /> with coefficients in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" />, the <a href="/wiki/Matrix_semiring" class="mw-redirect" title="Matrix semiring">matrix semiring</a> using ordinary <a href="/wiki/Matrix_addition" title="Matrix addition">addition</a> and <a href="/wiki/Matrix_multiplication" title="Matrix multiplication">multiplication</a> rules of matrices. Given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in {\mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in {\mathbb {N} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5d9ab68f55d202b14c6d41969d53ed7e245c47d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in {\mathbb {N} }}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" /> a semiring, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}}_{n}(R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}_{n}(R)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/898fd3711a7daf6e6998854deaf83fe985ffd896" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.582ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M}}_{n}(R)}" /> is always a semiring also. It is generally non-commutative even if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" /> was commutative. <a href="/wiki/Rng_(algebra)#Adjoining_an_identity_element_(Dorroh_extension)" title="Rng (algebra)">Dorroh extensions</a>: If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" /> is a semiring, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\times {\mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\times {\mathbb {N} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c19f6b70185f0b96bdb0571c4e492a2c313db9c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.282ex; height:2.176ex;" alt="{\displaystyle R\times {\mathbb {N} }}" /> with pointwise addition and multiplication given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,n\rangle \bullet \langle y,m\rangle :=\langle x\cdot y+(x\,m+y\,n),n\cdot m\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>n</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>∙</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>y</mi> <mo>,</mo> <mi>m</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>:=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>⋅</mo> <mi>y</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mi>m</mi> <mo>+</mo> <mi>y</mi> <mspace width="thinmathspace"></mspace> <mi>n</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>n</mi> <mo>⋅</mo> <mi>m</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,n\rangle \bullet \langle y,m\rangle :=\langle x\cdot y+(x\,m+y\,n),n\cdot m\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d9803e93b30e1e670e4ff6afa43b0b5752df465" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.853ex; height:2.843ex;" alt="{\displaystyle \langle x,n\rangle \bullet \langle y,m\rangle :=\langle x\cdot y+(x\,m+y\,n),n\cdot m\rangle }" /> defines another semiring with multiplicative unit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1_{R\times {\mathbb {N} }}:=\langle 0_{R},1_{\mathbb {N} }\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </mrow> </msub> <mo>:=</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo>,</mo> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1_{R\times {\mathbb {N} }}:=\langle 0_{R},1_{\mathbb {N} }\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c51a2cf018f4df148405e71166421f420e2b37f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.919ex; height:2.843ex;" alt="{\displaystyle 1_{R\times {\mathbb {N} }}:=\langle 0_{R},1_{\mathbb {N} }\rangle }" />. Very similarly, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" /> is any sub-semiring of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" />, one may also define a semiring on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\times N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>×</mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\times N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36596ebdbfdcf1f2c718c68885b4d3253659c1b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.668ex; height:2.176ex;" alt="{\displaystyle R\times N}" />, just by replacing the repeated addition in the formula by multiplication. Indeed, these constructions even work under looser conditions, as the structure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" /> is not actually required to have a multiplicative unit. <a href="/wiki/Zerosumfree_monoid" title="Zerosumfree monoid">Zerosumfree</a> semirings are in a sense furthest away from being rings. Given a semiring, one may adjoin a new zero <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>0</mn> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed8770da197ea2be2d3dce9fafad5b5c568e633d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.847ex; height:2.509ex;" alt="{\displaystyle 0'}" /> to the underlying set and thus obtain such a zerosumfree semiring that also lacks <a href="/wiki/Zero_divisor" title="Zero divisor">zero divisors</a>. In particular, now <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\cdot 0'=0'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>⋅</mo> <msup> <mn>0</mn> <mo>′</mo> </msup> <mo>=</mo> <msup> <mn>0</mn> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\cdot 0'=0'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/425fba97e9a48e5a1d55b63ec04508f349f40c76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.634ex; height:2.509ex;" alt="{\displaystyle 0\cdot 0'=0'}" /> and the old semiring is actually not a sub-semiring. One may then go on and adjoin new elements "on top" one at a time, while always respecting the zero. These two strategies also work under looser conditions. Sometimes the notations <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle -\infty }" /> resp. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bddbb0e4420a7e744cf71bd71216e11b0bf88831" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle +\infty }" /> are used when performing these constructions. Adjoining a new zero to the trivial semiring, in this way, results in another semiring which may be expressed in terms of the <a href="/wiki/Logical_connectives" class="mw-redirect" title="Logical connectives">logical connectives</a> of disjunction and conjunction: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \{0,1\},+,\cdot ,\langle 0,1\rangle \rangle =\langle \{\bot ,\top \},\lor ,\land ,\langle \bot ,\top \rangle \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo>+</mo> <mo>,</mo> <mo>⋅</mo> <mo>,</mo> <mo fence="false" stretchy="false">⟨</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mo fence="false" stretchy="false">{</mo> <mi mathvariant="normal">⊥</mi> <mo>,</mo> <mi mathvariant="normal">⊤</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo>∨</mo> <mo>,</mo> <mo>∧</mo> <mo>,</mo> <mo fence="false" stretchy="false">⟨</mo> <mi mathvariant="normal">⊥</mi> <mo>,</mo> <mi mathvariant="normal">⊤</mi> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \{0,1\},+,\cdot ,\langle 0,1\rangle \rangle =\langle \{\bot ,\top \},\lor ,\land ,\langle \bot ,\top \rangle \rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccf5da17647bec36d2584a6dcab6803084c2248c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.763ex; height:2.843ex;" alt="{\displaystyle \langle \{0,1\},+,\cdot ,\langle 0,1\rangle \rangle =\langle \{\bot ,\top \},\lor ,\land ,\langle \bot ,\top \rangle \rangle }" />. Consequently, this is the smallest semiring that is not a ring. Explicitly, it violates the ring axioms as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \top \lor P=\top }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">⊤</mi> <mo>∨</mo> <mi>P</mi> <mo>=</mo> <mi mathvariant="normal">⊤</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \top \lor P=\top }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9842a7d60da8435dfda5d3c009a4c3c043b0bca9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.043ex; height:2.176ex;" alt="{\displaystyle \top \lor P=\top }" /> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" />, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" /> has no additive inverse. In the <a href="/wiki/Duality_(order_theory)" title="Duality (order theory)">self-dual</a> definition, the fault is with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bot \land P=\bot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">⊥</mi> <mo>∧</mo> <mi>P</mi> <mo>=</mo> <mi mathvariant="normal">⊥</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bot \land P=\bot }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6573d505c26d7d3c76de3f0404611930ff46c40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.043ex; height:2.176ex;" alt="{\displaystyle \bot \land P=\bot }" />. (This is not to be conflated with the ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92aedfb5c02eff978ab963421ce930f46801657e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.605ex; height:2.509ex;" alt="{\displaystyle \mathbb {Z} _{2}}" />, whose addition functions as <a href="/wiki/Xor" class="mw-redirect" title="Xor">xor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \veebar }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊻</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \veebar }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04d326b0ae464569ba667f5b56e0541a4efe6bb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \veebar }" />.) In the <a href="/wiki/Set-theoretic_definition_of_natural_numbers" title="Set-theoretic definition of natural numbers">von Neumann model of the naturals</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0_{\omega }:=\{\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0_{\omega }:=\{\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a014303de47a66672c7c3325816eecbb89822271" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.487ex; height:2.843ex;" alt="{\displaystyle 0_{\omega }:=\{\}}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1_{\omega }:=\{0_{\omega }\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1_{\omega }:=\{0_{\omega }\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d59eae506b7aa4d0a704cb446dffcccee14c2ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.904ex; height:2.843ex;" alt="{\displaystyle 1_{\omega }:=\{0_{\omega }\}}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2_{\omega }:=\{0_{\omega },1_{\omega }\}={\mathcal {P}}1_{\omega }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> <mo>,</mo> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2_{\omega }:=\{0_{\omega },1_{\omega }\}={\mathcal {P}}1_{\omega }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dc0fb304f7eeac87cb8c147dbf238e855988a9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.574ex; height:2.843ex;" alt="{\displaystyle 2_{\omega }:=\{0_{\omega },1_{\omega }\}={\mathcal {P}}1_{\omega }}" />. The two-element semiring may be presented in terms of the set theoretic union and intersection as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle {\mathcal {P}}1_{\omega },\cup ,\cap ,\langle \{\},1_{\omega }\rangle \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> <mo>,</mo> <mo>∪</mo> <mo>,</mo> <mo>∩</mo> <mo>,</mo> <mo fence="false" stretchy="false">⟨</mo> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle {\mathcal {P}}1_{\omega },\cup ,\cap ,\langle \{\},1_{\omega }\rangle \rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7288d1ef2d360f5a791286d86b7ffa761bf3fcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.718ex; height:2.843ex;" alt="{\displaystyle \langle {\mathcal {P}}1_{\omega },\cup ,\cap ,\langle \{\},1_{\omega }\rangle \rangle }" />. Now this structure in fact still constitutes a semiring when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1_{\omega }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1_{\omega }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f067ccf0d2fc0104774ee03f815c8eecc8462e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.417ex; height:2.509ex;" alt="{\displaystyle 1_{\omega }}" /> is replaced by any inhabited set whatsoever. The <a href="/wiki/Ideal_(ring_theory)" title="Ideal (ring theory)">ideals</a> on a semiring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" />, with their standard operations on subset, form a lattice-ordered, simple and zerosumfree semiring. The ideals of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}}_{n}(R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}_{n}(R)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/898fd3711a7daf6e6998854deaf83fe985ffd896" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.582ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M}}_{n}(R)}" /> are in bijection with the ideals of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" />. The collection of left ideals of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" /> (and likewise the right ideals) also have much of that algebraic structure, except that then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" /> does not function as a two-sided multiplicative identity. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" /> is a semiring and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" /> is an <a href="/wiki/Inhabited_set" title="Inhabited set">inhabited set</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44e23745a51c2c2d8d91fd98c1cf721573747ece" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.797ex; height:2.343ex;" alt="{\displaystyle A^{*}}" /> denotes the <a href="/wiki/Free_monoid" title="Free monoid">free monoid</a> and the formal polynomials <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R[A^{*}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">[</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R[A^{*}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e4df514fa6d4e155cbcccc45f5e9882cbb4d247" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.855ex; height:2.843ex;" alt="{\displaystyle R[A^{*}]}" /> over its words form another semiring. For small sets, the generating elements are conventionally used to denote the polynomial semiring. For example, in case of a singleton <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=\{X\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>X</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\{X\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7794b18b5eea042a930fd6ecf37e3587d5b38a9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.146ex; height:2.843ex;" alt="{\displaystyle A=\{X\}}" /> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{*}=\{\varepsilon ,X,X^{2},X^{3},\dots \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>ε</mi> <mo>,</mo> <mi>X</mi> <mo>,</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> <mo>…</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{*}=\{\varepsilon ,X,X^{2},X^{3},\dots \}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40b393f8350363db314b0f562fced7d788938431" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.245ex; height:3.176ex;" alt="{\displaystyle A^{*}=\{\varepsilon ,X,X^{2},X^{3},\dots \}}" />, one writes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R[X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R[X]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afeccd52deabb878398a8485755c3ceea80caf9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.038ex; height:2.843ex;" alt="{\displaystyle R[X]}" />. Zerosumfree sub-semirings of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" /> can be used to determine sub-semirings of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R[A^{*}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">[</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R[A^{*}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e4df514fa6d4e155cbcccc45f5e9882cbb4d247" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.855ex; height:2.843ex;" alt="{\displaystyle R[A^{*}]}" />. Given a set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" />, not necessarily just a singleton, adjoining a default element to the set underlying a semiring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" /> one may define the semiring of partial functions from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" /> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" />. Given a <a href="/wiki/Derivation_(differential_algebra)" title="Derivation (differential algebra)">derivation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathrm {d} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathrm {d} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/610cb0f6ee5c12c4d8bb0aa39df669254e91ee78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle {\mathrm {d} }}" /> on a semiring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" />, another the operation "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bullet }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∙</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bullet }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3576c2406959ee194a6fc55c34b5ee9f6ffbb715" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \bullet }" />" fulfilling <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\bullet y=y\bullet X+{\mathrm {d} }(y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∙</mo> <mi>y</mi> <mo>=</mo> <mi>y</mi> <mo>∙</mo> <mi>X</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\bullet y=y\bullet X+{\mathrm {d} }(y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e32636ab53f58cca54647a90f234b14e36eb105a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.857ex; height:2.843ex;" alt="{\displaystyle X\bullet y=y\bullet X+{\mathrm {d} }(y)}" /> can be defined as part of a new multiplication on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R[X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R[X]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afeccd52deabb878398a8485755c3ceea80caf9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.038ex; height:2.843ex;" alt="{\displaystyle R[X]}" />, resulting in another semiring. The above is by no means an exhaustive list of systematic constructions. <div class="mw-heading mw-heading3"><h3 id="Derivations">Derivations</h3>[<a href="/w/index.php?title=Semiring&action=edit&section=5" title="Edit section: Derivations">edit</a>]</div> Derivations on a semiring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" /> are the maps <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathrm {d} }\colon R\to R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mo>:</mo> <mi>R</mi> <mo stretchy="false">→</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathrm {d} }\colon R\to R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c691b2d527e930ecfbbfa08984614572f3312004" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.469ex; height:2.176ex;" alt="{\displaystyle {\mathrm {d} }\colon R\to R}" /> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathrm {d} }(x+y)={\mathrm {d} }(x)+{\mathrm {d} }(y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathrm {d} }(x+y)={\mathrm {d} }(x)+{\mathrm {d} }(y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fdf189c2aaf953ace4f0fb26026572f69e5e8dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.055ex; height:2.843ex;" alt="{\displaystyle {\mathrm {d} }(x+y)={\mathrm {d} }(x)+{\mathrm {d} }(y)}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathrm {d} }(x\cdot y)={\mathrm {d} }(x)\cdot y+x\cdot {\mathrm {d} }(y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>⋅</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathrm {d} }(x\cdot y)={\mathrm {d} }(x)\cdot y+x\cdot {\mathrm {d} }(y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb586ab542e170ac399c9be288df7a62c75a571d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.737ex; height:2.843ex;" alt="{\displaystyle {\mathrm {d} }(x\cdot y)={\mathrm {d} }(x)\cdot y+x\cdot {\mathrm {d} }(y)}" />. For example, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" /> is the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\times 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\times 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8a0e3400ffb97d67c00267ed50cddfe824cbe80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 2\times 2}" /> unit matrix and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U={\bigl (}{\begin{smallmatrix}0&1\\0&0\end{smallmatrix}}{\bigr )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U={\bigl (}{\begin{smallmatrix}0&1\\0&0\end{smallmatrix}}{\bigr )}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bf5cc9a180c261a29ef006f4d56d60a7d9b987e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.18ex; height:3.343ex;" alt="{\displaystyle U={\bigl (}{\begin{smallmatrix}0&1\\0&0\end{smallmatrix}}{\bigr )}}" />, then the subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}}_{2}(R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}_{2}(R)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e411525b0304a4c36ca922f8a797dff7dda4e7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.418ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M}}_{2}(R)}" /> given by the matrices <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\,E+b\,U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mspace width="thinmathspace"></mspace> <mi>E</mi> <mo>+</mo> <mi>b</mi> <mspace width="thinmathspace"></mspace> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\,E+b\,U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72bd753935486e8846c784cfffccb990529e38b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.4ex; height:2.343ex;" alt="{\displaystyle a\,E+b\,U}" /> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b\in R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b\in R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d992800adce8f580e87a3c79b62f0e12d5349e1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.866ex; height:2.509ex;" alt="{\displaystyle a,b\in R}" /> is a semiring with derivation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\,E+b\,U\mapsto b\,U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mspace width="thinmathspace"></mspace> <mi>E</mi> <mo>+</mo> <mi>b</mi> <mspace width="thinmathspace"></mspace> <mi>U</mi> <mo stretchy="false">↦</mo> <mi>b</mi> <mspace width="thinmathspace"></mspace> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\,E+b\,U\mapsto b\,U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3d04d9a5942b443eecd8c1511aab61141d43cb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.181ex; height:2.343ex;" alt="{\displaystyle a\,E+b\,U\mapsto b\,U}" />. <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2>[<a href="/w/index.php?title=Semiring&action=edit&section=6" title="Edit section: Properties">edit</a>]</div> A basic property of semirings is that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" /> is not a left or right <a href="/wiki/Zero_divisor" title="Zero divisor">zero divisor</a>, and that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" /> but also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" /> squares to itself, i.e. these have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u^{2}=u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u^{2}=u}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91770b992e768b841db82546ffea6e2eb9bfe504" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.812ex; height:2.676ex;" alt="{\displaystyle u^{2}=u}" />. Some notable properties are inherited from the monoid structures: The monoid axioms demand unit existence, and so the set underlying a semiring cannot be empty. Also, the <a href="/wiki/Arity" title="Arity">2-ary</a> predicate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\leq _{\text{pre}}y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <msub> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>pre</mtext> </mrow> </msub> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\leq _{\text{pre}}y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/336f6f0f2140f8b6abcc5a0965421342b60bd78e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.104ex; height:2.676ex;" alt="{\displaystyle x\leq _{\text{pre}}y}" /> defined as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists d.x+d=y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mi>d</mi> <mo>.</mo> <mi>x</mi> <mo>+</mo> <mi>d</mi> <mo>=</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists d.x+d=y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7eb6bcc5e50a7615fb98ffe1be76c0be8decc87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.182ex; height:2.509ex;" alt="{\displaystyle \exists d.x+d=y}" />, here defined for the addition operation, always constitutes the right <a href="/wiki/Monoid#Commutative_monoid" title="Monoid">canonical</a> <a href="/wiki/Preorder" title="Preorder">preorder</a> relation. <a href="/wiki/Reflexive_relation" title="Reflexive relation">Reflexivity</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\leq _{\text{pre}}y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <msub> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>pre</mtext> </mrow> </msub> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\leq _{\text{pre}}y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfedff93018d27c81d5488e85273a9d6d6647ba6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.93ex; height:2.676ex;" alt="{\displaystyle y\leq _{\text{pre}}y}" /> is witnessed by the identity. Further, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq _{\text{pre}}y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <msub> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>pre</mtext> </mrow> </msub> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq _{\text{pre}}y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c66a5ce76ecc4944a1f32abb8559001204ca3f19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.937ex; height:2.843ex;" alt="{\displaystyle 0\leq _{\text{pre}}y}" /> is always valid, and so zero is the <a href="/wiki/Least_element" class="mw-redirect" title="Least element">least element</a> with respect to this preorder. Considering it for the commutative addition in particular, the distinction of "right" may be disregarded. In the non-negative integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }" />, for example, this relation is <a href="/wiki/Anti-symmetric_relation" class="mw-redirect" title="Anti-symmetric relation">anti-symmetric</a> and <a href="/wiki/Strongly_connected" class="mw-redirect" title="Strongly connected">strongly connected</a>, and thus in fact a (non-strict) <a href="/wiki/Total_order" title="Total order">total order</a>. Below, more conditional properties are discussed. <div class="mw-heading mw-heading3"><h3 id="Semifields">Semifields</h3>[<a href="/w/index.php?title=Semiring&action=edit&section=7" title="Edit section: Semifields">edit</a>]</div> Any <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> is also a <a href="/wiki/Semifield" title="Semifield">semifield</a>, which in turn is a semiring in which also multiplicative inverses exist. <div class="mw-heading mw-heading3"><h3 id="Rings">Rings</h3>[<a href="/w/index.php?title=Semiring&action=edit&section=8" title="Edit section: Rings">edit</a>]</div> Any field is also a ring, which in turn is a semiring in which also additive inverses exist. Note that a semiring omits such a requirement, i.e., it requires only a <a href="/wiki/Commutative_monoid" class="mw-redirect" title="Commutative monoid">commutative monoid</a>, not a <a href="/wiki/Commutative_group" class="mw-redirect" title="Commutative group">commutative group</a>. The extra requirement for a ring itself already implies the existence of a multiplicative zero. This contrast is also why for the theory of semirings, the multiplicative zero must be specified explicitly. Here <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704fb0427140d054dd267925495e78164fee9aac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}" />, the additive inverse of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" />, squares to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" />. As additive differences <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d=y-x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <mi>y</mi> <mo>−</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d=y-x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/427f5425a249100b4c24b5519f3d74d03dfa0db1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.64ex; height:2.509ex;" alt="{\displaystyle d=y-x}" /> always exist in a ring, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\leq _{\text{pre}}y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <msub> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>pre</mtext> </mrow> </msub> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\leq _{\text{pre}}y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/336f6f0f2140f8b6abcc5a0965421342b60bd78e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.104ex; height:2.676ex;" alt="{\displaystyle x\leq _{\text{pre}}y}" /> is a trivial binary relation in a ring. <div class="mw-heading mw-heading3"><h3 id="Commutative_semirings">Commutative semirings</h3>[<a href="/w/index.php?title=Semiring&action=edit&section=9" title="Edit section: Commutative semirings">edit</a>]</div> A semiring is called a <a href="/wiki/Commutative" class="mw-redirect" title="Commutative">commutative</a> semiring if also the multiplication is commutative.<a href="#cite_note-FOOTNOTELothaire2005212-9">[8]</a> Its axioms can be stated concisely: It consists of two commutative monoids <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle +,0\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mo>+</mo> <mo>,</mo> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle +,0\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fb339560c7c863582af8512618ed6ffa4f5d3ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.814ex; height:2.843ex;" alt="{\displaystyle \langle +,0\rangle }" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \cdot ,1\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mo>⋅</mo> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \cdot ,1\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95fbd3f5ebdfc68306751cdfe6dd319cd34a5dde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle \langle \cdot ,1\rangle }" /> on one set such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot 0=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mn>0</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot 0=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31f39efe37450f6bce024486f458a90d11a08278" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.332ex; height:2.176ex;" alt="{\displaystyle a\cdot 0=0}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot (b+c)=a\cdot b+a\cdot c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo>+</mo> <mi>a</mi> <mo>⋅</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot (b+c)=a\cdot b+a\cdot c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8827e12f09f1ab8a5f3d7783b7357bd4cc398db7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.324ex; height:2.843ex;" alt="{\displaystyle a\cdot (b+c)=a\cdot b+a\cdot c}" />. The <a href="/wiki/Center_(ring_theory)" title="Center (ring theory)">center</a> of a semiring is a sub-semiring and being commutative is equivalent to being its own center. The commutative semiring of natural numbers is the <a href="/wiki/Initial_object" class="mw-redirect" title="Initial object">initial object</a> among its kind, meaning there is a unique structure preserving map of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbb {N} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bbdf4a20a0a0a50daaa524863051d5d22c9fb23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle {\mathbb {N} }}" /> into any commutative semiring. The bounded distributive lattices are <a href="/wiki/Partial_order" class="mw-redirect" title="Partial order">partially ordered</a>, commutative semirings fulfilling certain algebraic equations relating to distributivity and idempotence. Thus so are their <a href="/wiki/Duality_theory_for_distributive_lattices" title="Duality theory for distributive lattices">duals</a>. <div class="mw-heading mw-heading3"><h3 id="Ordered_semirings">Ordered semirings</h3>[<a href="/w/index.php?title=Semiring&action=edit&section=10" title="Edit section: Ordered semirings">edit</a>]</div> Notions or order can be defined using strict, non-strict or <a href="/wiki/Second-order_logic" title="Second-order logic">second-order</a> formulations. Additional properties such as commutativity simplify the axioms. Given a <a href="/wiki/Strict_total_order" class="mw-redirect" title="Strict total order">strict total order</a> (also sometimes called linear order, or <a href="/wiki/Pseudo-order" title="Pseudo-order">pseudo-order</a> in a constructive formulation), then by definition, the positive and negative elements fulfill <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35b0ac4e2c02e608588f1525eac793c53eb7c55f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle 0<x}" /> resp. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x<0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a4dbbf970b2d2863dcab589eafe006f08e727d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x<0}" />. By irreflexivity of a strict order, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" /> is a left zero divisor, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\cdot x<s\cdot y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>⋅</mo> <mi>x</mi> <mo><</mo> <mi>s</mi> <mo>⋅</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s\cdot x<s\cdot y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d721c2813543a2f3490d15a7b59c12afcbeb0cd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.123ex; height:2.176ex;" alt="{\displaystyle s\cdot x<s\cdot y}" /> is false. The non-negative elements are characterized by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg (x<0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo><</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg (x<0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fd53715a16e96928a4576935546af26935a090d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.95ex; height:2.843ex;" alt="{\displaystyle \neg (x<0)}" />, which is then written <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3744ec5c328cae43887a6ed07a7a9c388634758c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.591ex; height:2.343ex;" alt="{\displaystyle 0\leq x}" />. Generally, the strict total order can be negated to define an associated partial order. The <a href="/wiki/Asymmetric_relation" title="Asymmetric relation">asymmetry</a> of the former manifests as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x<y\to x\leq y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo><</mo> <mi>y</mi> <mo stretchy="false">→</mo> <mi>x</mi> <mo>≤</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x<y\to x\leq y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fda46046df8b528c3ddb2841b49b59ff01641ba7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.781ex; height:2.343ex;" alt="{\displaystyle x<y\to x\leq y}" />. In fact in <a href="/wiki/Classical_logic" title="Classical logic">classical mathematics</a> the latter is a (non-strict) total order and such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3744ec5c328cae43887a6ed07a7a9c388634758c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.591ex; height:2.343ex;" alt="{\displaystyle 0\leq x}" /> implies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0\lor 0<x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mo>∨</mo> <mn>0</mn> <mo><</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=0\lor 0<x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95ec1e17db265b9e6bbac89b87a3a7f3e263ca56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.764ex; height:2.176ex;" alt="{\displaystyle x=0\lor 0<x}" />. Likewise, given any (non-strict) total order, its negation is <a href="/wiki/Irreflexive_relation" class="mw-redirect" title="Irreflexive relation">irreflexive</a> and <a href="/wiki/Transitive_relation" title="Transitive relation">transitive</a>, and those two properties found together are sometimes called strict quasi-order. Classically this defines a strict total order – indeed strict total order and total order can there be defined in terms of one another. Recall that "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leq _{\text{pre}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>pre</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leq _{\text{pre}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50b73be8fc870461f236e6de735bb7611166fa1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.329ex; height:2.676ex;" alt="{\displaystyle \leq _{\text{pre}}}" />" defined above is trivial in any ring. The existence of rings that admit a non-trivial non-strict order shows that these need not necessarily coincide with "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leq _{\text{pre}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>pre</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leq _{\text{pre}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50b73be8fc870461f236e6de735bb7611166fa1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.329ex; height:2.676ex;" alt="{\displaystyle \leq _{\text{pre}}}" />". <div class="mw-heading mw-heading4"><h4 id="Additively_idempotent_semirings">Additively idempotent semirings</h4>[<a href="/w/index.php?title=Semiring&action=edit&section=11" title="Edit section: Additively idempotent semirings">edit</a>]</div> A semiring in which every element is an additive <a href="/wiki/Idempotent" class="mw-redirect" title="Idempotent">idempotent</a>, that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x+x=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>+</mo> <mi>x</mi> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x+x=x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e003cc73ff9f8bfc9f7f959b64662b9e15ce4fc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.928ex; height:2.176ex;" alt="{\displaystyle x+x=x}" /> for all elements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" />, is called an (additively) idempotent semiring.<a href="#cite_note-Esik08-10">[9]</a> Establishing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+1=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+1=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5120dbef0aa4b1760be59a33516661780dfc7a14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.426ex; height:2.343ex;" alt="{\displaystyle 1+1=1}" /> suffices. Be aware that sometimes this is just called idempotent semiring, regardless of rules for multiplication. In such a semiring, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\leq _{\text{pre}}y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <msub> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>pre</mtext> </mrow> </msub> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\leq _{\text{pre}}y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/336f6f0f2140f8b6abcc5a0965421342b60bd78e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.104ex; height:2.676ex;" alt="{\displaystyle x\leq _{\text{pre}}y}" /> is equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x+y=y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>=</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x+y=y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cd28a38f534feae59f3224c0ef7c1e4e7efb18d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.58ex; height:2.343ex;" alt="{\displaystyle x+y=y}" /> and always constitutes a partial order, here now denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\leq y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≤</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\leq y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c07a0bc023490be1c08e6c33a9cdc93bec908224" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.343ex;" alt="{\displaystyle x\leq y}" />. In particular, here <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\leq 0\leftrightarrow x=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≤</mo> <mn>0</mn> <mo stretchy="false">↔</mo> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\leq 0\leftrightarrow x=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be776066cdac09292512eeffccf459a4f74bae75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.795ex; height:2.343ex;" alt="{\displaystyle x\leq 0\leftrightarrow x=0}" />. So additively idempotent semirings are zerosumfree and, indeed, the only additively idempotent semiring that has all additive inverses is the trivial ring and so this property is specific to semiring theory. Addition and multiplication respect the ordering in the sense that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\leq y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≤</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\leq y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c07a0bc023490be1c08e6c33a9cdc93bec908224" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.343ex;" alt="{\displaystyle x\leq y}" /> implies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x+t\leq y+t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>+</mo> <mi>t</mi> <mo>≤</mo> <mi>y</mi> <mo>+</mo> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x+t\leq y+t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d271e43a1553cc78b9b897e31a7fa2b5885449c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.944ex; height:2.343ex;" alt="{\displaystyle x+t\leq y+t}" />, and furthermore implies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\cdot x\leq s\cdot y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>⋅</mo> <mi>x</mi> <mo>≤</mo> <mi>s</mi> <mo>⋅</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s\cdot x\leq s\cdot y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c98cb6d8b4d2ef87c93b9fb8aba9de4066892a34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.123ex; height:2.343ex;" alt="{\displaystyle s\cdot x\leq s\cdot y}" /> as well as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\cdot s\leq y\cdot s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>⋅</mo> <mi>s</mi> <mo>≤</mo> <mi>y</mi> <mo>⋅</mo> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\cdot s\leq y\cdot s}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bdb2985f7798fea416a7d718baf56ae64f23de2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.123ex; height:2.343ex;" alt="{\displaystyle x\cdot s\leq y\cdot s}" />, for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y,t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y,t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77a04fe169d6625eedb4e4db82339863f01f5ef9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.393ex; height:2.343ex;" alt="{\displaystyle x,y,t}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" />. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" /> is additively idempotent, then so are the polynomials in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R[X^{*}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">[</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R[X^{*}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/703fd4f04d719ca84165f4b11d57966b16cbfb5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.109ex; height:2.843ex;" alt="{\displaystyle R[X^{*}]}" />. A semiring such that there is a lattice structure on its underlying set is lattice-ordered if the sum coincides with the meet, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x+y=x\lor y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>=</mo> <mi>x</mi> <mo>∨</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x+y=x\lor y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cfde186895f39936c5ea731329de673f96ed901" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.492ex; height:2.343ex;" alt="{\displaystyle x+y=x\lor y}" />, and the product lies beneath the join <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\cdot y\leq x\land y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>⋅</mo> <mi>y</mi> <mo>≤</mo> <mi>x</mi> <mo>∧</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\cdot y\leq x\land y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5508ad6785757d16f1dd5be660e3647d99e0c2ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.33ex; height:2.343ex;" alt="{\displaystyle x\cdot y\leq x\land y}" />. The lattice-ordered semiring of ideals on a semiring is not necessarily <a href="/wiki/Distributive_lattice" title="Distributive lattice">distributive with respect to</a> the lattice structure. More strictly than just additive idempotence, a semiring is called simple iff <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x+1=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x+1=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be7d8629365fe7f585b719f0a76330a2c358c6f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.593ex; height:2.343ex;" alt="{\displaystyle x+1=1}" /> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" />. Then also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+1=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+1=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5120dbef0aa4b1760be59a33516661780dfc7a14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.426ex; height:2.343ex;" alt="{\displaystyle 1+1=1}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\leq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≤</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\leq 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97bc1f8c61d5129d13278e6ee53069a1736ce3c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.591ex; height:2.343ex;" alt="{\displaystyle x\leq 1}" /> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" />. Here <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" /> then functions akin to an additively infinite element. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" /> is an additively idempotent semiring, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x\in R\mid x+1=1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mi>R</mi> <mo>∣</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{x\in R\mid x+1=1\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85532fce34a4e1f29b1c426f84d9eb06ef70c945" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.79ex; height:2.843ex;" alt="{\displaystyle \{x\in R\mid x+1=1\}}" /> with the inherited operations is its simple sub-semiring. An example of an additively idempotent semiring that is not simple is the <a href="/wiki/Tropical_semiring" title="Tropical semiring">tropical semiring</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbb {R} }\cup \{-\infty \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbb {R} }\cup \{-\infty \}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a69d5077221f05902cdd81154bfba38736f9e58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.717ex; height:2.843ex;" alt="{\displaystyle {\mathbb {R} }\cup \{-\infty \}}" /> with the 2-ary maximum function, with respect to the standard order, as addition. Its simple sub-semiring is trivial. A c-semiring is an idempotent semiring and with addition defined over arbitrary sets. An additively idempotent semiring with idempotent multiplication, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}=x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b995c9f99ce207b372a7f9446f59e02ce7934544" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.812ex; height:2.676ex;" alt="{\displaystyle x^{2}=x}" />, is called additively and multiplicatively idempotent semiring, but sometimes also just idempotent semiring. The commutative, simple semirings with that property are exactly the bounded distributive lattices with unique minimal and maximal element (which then are the units). <a href="/wiki/Heyting_algebras" class="mw-redirect" title="Heyting algebras">Heyting algebras</a> are such semirings and the <a href="/wiki/Boolean_algebra_(structure)" title="Boolean algebra (structure)">Boolean algebras</a> are a special case. Further, given two bounded distributive lattices, there are constructions resulting in commutative additively-idempotent semirings, which are more complicated than just the direct sum of structures. <div class="mw-heading mw-heading4"><h4 id="Number_lines">Number lines</h4>[<a href="/w/index.php?title=Semiring&action=edit&section=12" title="Edit section: Number lines">edit</a>]</div> In a model of the ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbb {R} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbb {R} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c35d4fb24ff006be5c264f4a3cf7760653a06b30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle {\mathbb {R} }}" />, one can define a non-trivial positivity predicate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35b0ac4e2c02e608588f1525eac793c53eb7c55f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle 0<x}" /> and a predicate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x<y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo><</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x<y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aeb239de6fee56ea8b6a65f7858d95b87632069f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.176ex;" alt="{\displaystyle x<y}" /> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<(y-x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<(y-x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4577370c591cca1d82dbb3304dcf1a17a63eadfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.396ex; height:2.843ex;" alt="{\displaystyle 0<(y-x)}" /> that constitutes a strict total order, which fulfills properties such as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg (x<0\lor 0<x)\to x=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo><</mo> <mn>0</mn> <mo>∨</mo> <mn>0</mn> <mo><</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg (x<0\lor 0<x)\to x=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82651758fe5e101555c1edab5cf59857e19e6863" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.328ex; height:2.843ex;" alt="{\displaystyle \neg (x<0\lor 0<x)\to x=0}" />, or classically the <a href="/wiki/Law_of_trichotomy" title="Law of trichotomy">law of trichotomy</a>. With its standard addition and multiplication, this structure forms the strictly <a href="/wiki/Ordered_field" title="Ordered field">ordered field</a> that is <a href="/wiki/Dedekind-complete" class="mw-redirect" title="Dedekind-complete">Dedekind-complete</a>. <a href="/wiki/Elementary_equivalence" title="Elementary equivalence">By definition</a>, all <a href="/wiki/First-order_logic" title="First-order logic">first-order properties</a> proven in the theory of the reals are also provable in the <a href="/wiki/Decidability_(logic)#Decidability_of_a_theory" title="Decidability (logic)">decidable theory</a> of the <a href="/wiki/Real_closed_field" title="Real closed field">real closed field</a>. For example, here <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x<y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo><</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x<y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aeb239de6fee56ea8b6a65f7858d95b87632069f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.176ex;" alt="{\displaystyle x<y}" /> is mutually exclusive with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists d.y+d^{2}=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mi>d</mi> <mo>.</mo> <mi>y</mi> <mo>+</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists d.y+d^{2}=x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8888578f84e90bb7215c2a6ab5e457bd3f37eba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.239ex; height:3.009ex;" alt="{\displaystyle \exists d.y+d^{2}=x}" />. But beyond just ordered fields, the four properties listed below are also still valid in many sub-semirings of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbb {R} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbb {R} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c35d4fb24ff006be5c264f4a3cf7760653a06b30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle {\mathbb {R} }}" />, including the rationals, the integers, as well as the non-negative parts of each of these structures. In particular, the non-negative reals, the non-negative rationals and the non-negative integers are such a semirings. The first two properties are analogous to the property valid in the idempotent semirings: Translation and scaling respect these <a href="/wiki/Ordered_ring" title="Ordered ring">ordered rings</a>, in the sense that addition and multiplication in this ring validate <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x<y)\,\to \,x+t<y+t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo><</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mo stretchy="false">→</mo> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mi>t</mi> <mo><</mo> <mi>y</mi> <mo>+</mo> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x<y)\,\to \,x+t<y+t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf7e6fa310589d70c69fd780120125a10955ea01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.725ex; height:2.843ex;" alt="{\displaystyle (x<y)\,\to \,x+t<y+t}" /></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x<y\land 0<s)\,\to \,s\cdot x<s\cdot y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo><</mo> <mi>y</mi> <mo>∧</mo> <mn>0</mn> <mo><</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mo stretchy="false">→</mo> <mspace width="thinmathspace"></mspace> <mi>s</mi> <mo>⋅</mo> <mi>x</mi> <mo><</mo> <mi>s</mi> <mo>⋅</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x<y\land 0<s)\,\to \,s\cdot x<s\cdot y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a06b3a6a9fc2cba2d89369b1dcc2643bccf2dbef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.838ex; height:2.843ex;" alt="{\displaystyle (x<y\land 0<s)\,\to \,s\cdot x<s\cdot y}" /></li></ul> In particular, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0<y\land 0<s)\to 0<s\cdot y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo><</mo> <mi>y</mi> <mo>∧</mo> <mn>0</mn> <mo><</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mn>0</mn> <mo><</mo> <mi>s</mi> <mo>⋅</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0<y\land 0<s)\to 0<s\cdot y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/643c6053c9c4245b8efa430c55bd6378211426f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.96ex; height:2.843ex;" alt="{\displaystyle (0<y\land 0<s)\to 0<s\cdot y}" /> and so squaring of elements preserves positivity. Take note of two more properties that are always valid in a ring. Firstly, trivially <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\,\to \,x\leq _{\text{pre}}y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mspace width="thinmathspace"></mspace> <mo stretchy="false">→</mo> <mspace width="thinmathspace"></mspace> <mi>x</mi> <msub> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>pre</mtext> </mrow> </msub> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\,\to \,x\leq _{\text{pre}}y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4628d4bcb2681b1dc9f80909e13d24c13a6314db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.238ex; height:2.843ex;" alt="{\displaystyle P\,\to \,x\leq _{\text{pre}}y}" /> for any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" />. In particular, the positive additive difference existence can be expressed as <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x<y)\,\to \,x\leq _{\text{pre}}y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo><</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mo stretchy="false">→</mo> <mspace width="thinmathspace"></mspace> <mi>x</mi> <msub> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>pre</mtext> </mrow> </msub> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x<y)\,\to \,x\leq _{\text{pre}}y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d2a1687a0413ea7cf21878e18a512710e2ddb5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.886ex; height:3.009ex;" alt="{\displaystyle (x<y)\,\to \,x\leq _{\text{pre}}y}" /></li></ul> Secondly, in the presence of a trichotomous order, the non-zero elements of the additive group are partitioned into positive and negative elements, with the inversion operation moving between them. With <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-1)^{2}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-1)^{2}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b0dc632f669ba5bebd0626878471178e3e981c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.095ex; height:3.176ex;" alt="{\displaystyle (-1)^{2}=1}" />, all squares are proven non-negative. Consequently, non-trivial rings have a positive multiplicative unit, <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b33e907a155a8f183638e65efa7465ff8c47335f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.423ex; height:2.176ex;" alt="{\displaystyle 0<1}" /></li></ul> Having discussed a strict order, it follows that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\neq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≠</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\neq 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5add705e86314a6ce57c76d7493896b092661a75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.423ex; height:2.676ex;" alt="{\displaystyle 0\neq 1}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\neq 1+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≠</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\neq 1+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09c70c53e536e8ded455c8805e63ab2ef761891a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.426ex; height:2.676ex;" alt="{\displaystyle 1\neq 1+1}" />, etc. <div class="mw-heading mw-heading4"><h4 id="Discretely_ordered_semirings">Discretely ordered semirings</h4>[<a href="/w/index.php?title=Semiring&action=edit&section=13" title="Edit section: Discretely ordered semirings">edit</a>]</div> There are a few conflicting notions of discreteness in order theory. Given some strict order on a semiring, one such notion is given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" /> being positive and <a href="/wiki/Covering_relation" title="Covering relation">covering</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" />, i.e. there being no element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" /> between the units, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg (0<x\land x<1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo><</mo> <mi>x</mi> <mo>∧</mo> <mi>x</mi> <mo><</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg (0<x\land x<1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cea9a884a37335ed338af71dda8677a6d97f05b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.123ex; height:2.843ex;" alt="{\displaystyle \neg (0<x\land x<1)}" />. Now in the present context, an order shall be called discrete if this is fulfilled and, furthermore, all elements of the semiring are non-negative, so that the semiring starts out with the units. Denote by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {PA}}^{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {PA}}^{-}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/798737a1975f160943f3b8bef6bb1d5ff72a4779" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.546ex; height:2.509ex;" alt="{\displaystyle {\mathsf {PA}}^{-}}" /> the theory of a commutative, discretely ordered semiring also validating the above four properties relating a strict order with the algebraic structure. All of its models have the model <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }" /> as its initial segment and <a href="/wiki/G%C3%B6del%27s_theorems" class="mw-redirect" title="Gödel's theorems">Gödel incompleteness</a> and <a href="/wiki/Tarski_undefinability_theorem" class="mw-redirect" title="Tarski undefinability theorem">Tarski undefinability</a> already apply to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {PA}}^{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {PA}}^{-}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/798737a1975f160943f3b8bef6bb1d5ff72a4779" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.546ex; height:2.509ex;" alt="{\displaystyle {\mathsf {PA}}^{-}}" />. The non-negative elements of a commutative, <a href="/wiki/Ordered_ring#Discrete_ordered_rings" title="Ordered ring">discretely ordered ring</a> always validate the axioms of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {PA}}^{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {PA}}^{-}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/798737a1975f160943f3b8bef6bb1d5ff72a4779" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.546ex; height:2.509ex;" alt="{\displaystyle {\mathsf {PA}}^{-}}" />. So a slightly more exotic model of the theory is given by the positive elements in the <a href="/wiki/Polynomial_ring" title="Polynomial ring">polynomial ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbb {Z} }[X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbb {Z} }[X]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/410cd3eb154d84dc78cd92f1129e96e399e61a7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.824ex; height:2.843ex;" alt="{\displaystyle {\mathbb {Z} }[X]}" />, with positivity predicate for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p={\textstyle \sum }_{k=0}^{n}a_{k}X^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>∑</mo> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p={\textstyle \sum }_{k=0}^{n}a_{k}X^{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/208b1e9f6d7a0700739123950dbea88bc7e2937b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:15.404ex; height:3.343ex;" alt="{\displaystyle p={\textstyle \sum }_{k=0}^{n}a_{k}X^{k}}" /> defined in terms of the last non-zero coefficient, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<p:=(0<a_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>p</mi> <mo>:=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo><</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<p:=(0<a_{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e056cf2554c8cec6614e1c323fabd2672b18e843" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.694ex; height:2.843ex;" alt="{\displaystyle 0<p:=(0<a_{n})}" />, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p<q:=(0<q-p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo><</mo> <mi>q</mi> <mo>:=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo><</mo> <mi>q</mi> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p<q:=(0<q-p)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a551cd8ee6e9ddc5c8028a5eec712c999fff1c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:20.322ex; height:2.843ex;" alt="{\displaystyle p<q:=(0<q-p)}" /> as above. While <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {PA}}^{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {PA}}^{-}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/798737a1975f160943f3b8bef6bb1d5ff72a4779" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.546ex; height:2.509ex;" alt="{\displaystyle {\mathsf {PA}}^{-}}" /> proves all <a href="/wiki/Arithmetical_hierarchy" title="Arithmetical hierarchy"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma _{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d37ee4ac26b75c4bca37e546fcc9859f3e45adf9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.732ex; height:2.509ex;" alt="{\displaystyle \Sigma _{1}}" />-sentences</a> that are true about <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }" />, beyond this complexity one can find simple such statements that are <a href="/wiki/Logical_independence" class="mw-redirect" title="Logical independence">independent</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {PA}}^{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {PA}}^{-}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/798737a1975f160943f3b8bef6bb1d5ff72a4779" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.546ex; height:2.509ex;" alt="{\displaystyle {\mathsf {PA}}^{-}}" />. For example, while <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi _{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4aab5a28da997de9084eef3e569bd1e072efc1aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle \Pi _{1}}" />-sentences true about <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }" /> are still true for the other model just defined, inspection of the polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" /> demonstrates <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {PA}}^{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {PA}}^{-}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/798737a1975f160943f3b8bef6bb1d5ff72a4779" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.546ex; height:2.509ex;" alt="{\displaystyle {\mathsf {PA}}^{-}}" />-independence of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi _{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10e0f32ab9da5560199913701cfdb210e7b32736" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle \Pi _{2}}" />-claim that all numbers are of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f55d51e4d55cdb8b9bd331806a8f49860597a7fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.232ex; height:2.509ex;" alt="{\displaystyle 2q}" /> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2q+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>q</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2q+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6889cb7ac4a0654678b4d65b8b4b25d40019f5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.235ex; height:2.509ex;" alt="{\displaystyle 2q+1}" /> ("<a href="/wiki/Parity_(mathematics)#Definition" title="Parity (mathematics)">odd or even</a>"). Showing that also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbb {Z} }[X,Y]/(X^{2}-2Y^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <msup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbb {Z} }[X,Y]/(X^{2}-2Y^{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd7fdccf46cf82403bd1cda2f259360e519e4cab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.611ex; height:3.176ex;" alt="{\displaystyle {\mathbb {Z} }[X,Y]/(X^{2}-2Y^{2})}" /> can be discretely ordered demonstrates that the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi _{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4aab5a28da997de9084eef3e569bd1e072efc1aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle \Pi _{1}}" />-claim <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}\neq 2y^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>≠</mo> <mn>2</mn> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}\neq 2y^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8443f0b7d68ebdd0c960c286c016f19a16835944" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.86ex; height:3.176ex;" alt="{\displaystyle x^{2}\neq 2y^{2}}" /> for non-zero <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" /> ("no rational squared equals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}" />") is independent. Likewise, analysis for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbb {Z} }[X,Y,Z]/(XZ-Y^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo>,</mo> <mi>Z</mi> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mi>Z</mi> <mo>−</mo> <msup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbb {Z} }[X,Y,Z]/(XZ-Y^{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e330d97dab88c3475e903ab3a7aac05db6ae1d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.773ex; height:3.176ex;" alt="{\displaystyle {\mathbb {Z} }[X,Y,Z]/(XZ-Y^{2})}" /> demonstrates independence of some statements about <a href="/wiki/Factorization" title="Factorization">factorization</a> true in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }" />. There are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {PA}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {PA}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6812979524cbdd07e79311aa1263ab6db58758f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.036ex; height:2.176ex;" alt="{\displaystyle {\mathsf {PA}}}" /> characterizations of primality that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {PA}}^{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {PA}}^{-}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/798737a1975f160943f3b8bef6bb1d5ff72a4779" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.546ex; height:2.509ex;" alt="{\displaystyle {\mathsf {PA}}^{-}}" /> does not validate for the number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}" />. In the other direction, from any model of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {PA}}^{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {PA}}^{-}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/798737a1975f160943f3b8bef6bb1d5ff72a4779" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.546ex; height:2.509ex;" alt="{\displaystyle {\mathsf {PA}}^{-}}" /> one may construct an ordered ring, which then has elements that are negative with respect to the order, that is still discrete the sense that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" /> covers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" />. To this end one defines an equivalence class of pairs from the original semiring. Roughly, the ring corresponds to the differences of elements in the old structure, generalizing the way in which the <a href="/wiki/Initial_object" class="mw-redirect" title="Initial object">initial</a> ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" /> <a href="/wiki/Integer#Equivalence_classes_of_ordered_pairs" title="Integer">can be defined from</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }" />. This, in effect, adds all the inverses and then the preorder is again trivial in that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x.x\leq _{\text{pre}}0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>.</mo> <mi>x</mi> <msub> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>pre</mtext> </mrow> </msub> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x.x\leq _{\text{pre}}0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa43e710ac87444d8d9e2a3d883eacfae0468c53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.768ex; height:2.843ex;" alt="{\displaystyle \forall x.x\leq _{\text{pre}}0}" />. Beyond the size of the two-element algebra, no simple semiring starts out with the units. Being discretely ordered also stands in contrast to, e.g., the standard ordering on the semiring of non-negative rationals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbb {Q} }_{\geq 0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>≥</mo> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbb {Q} }_{\geq 0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1dca8c755de18b7a703ad7272a179ac0617f1e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.141ex; height:2.843ex;" alt="{\displaystyle {\mathbb {Q} }_{\geq 0}}" />, which is <a href="/wiki/Dense_order" title="Dense order">dense</a> between the units. For another example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbb {Z} }[X]/(2X^{2}-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbb {Z} }[X]/(2X^{2}-1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef9bc6ad9177d0ef1629dafe52f2e75f89df39ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.012ex; height:3.176ex;" alt="{\displaystyle {\mathbb {Z} }[X]/(2X^{2}-1)}" /> can be ordered, but not discretely so. <div class="mw-heading mw-heading4"><h4 id="Natural_numbers">Natural numbers</h4>[<a href="/w/index.php?title=Semiring&action=edit&section=14" title="Edit section: Natural numbers">edit</a>]</div> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {PA}}^{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {PA}}^{-}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/798737a1975f160943f3b8bef6bb1d5ff72a4779" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.546ex; height:2.509ex;" alt="{\displaystyle {\mathsf {PA}}^{-}}" /> plus <a href="/wiki/Mathematical_induction" title="Mathematical induction">mathematical induction</a> gives <a href="/wiki/Peano_axioms#Equivalent_axiomatizations" title="Peano axioms">a theory equivalent to</a> first-order <a href="/wiki/Peano_arithmetic" class="mw-redirect" title="Peano arithmetic">Peano arithmetic</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {PA}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {PA}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6812979524cbdd07e79311aa1263ab6db58758f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.036ex; height:2.176ex;" alt="{\displaystyle {\mathsf {PA}}}" />. The theory is also famously not <a href="/wiki/Categorical_theory" title="Categorical theory">categorical</a>, but <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }" /> is of course the intended model. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {PA}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {PA}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6812979524cbdd07e79311aa1263ab6db58758f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.036ex; height:2.176ex;" alt="{\displaystyle {\mathsf {PA}}}" /> proves that there are no zero divisors and it is zerosumfree and so no <a href="/wiki/Non-standard_model_of_arithmetic" title="Non-standard model of arithmetic">model of it</a> is a ring. The standard axiomatization of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {PA}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {PA}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6812979524cbdd07e79311aa1263ab6db58758f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.036ex; height:2.176ex;" alt="{\displaystyle {\mathsf {PA}}}" /> is more concise and the theory of its order is commonly treated in terms of the non-strict "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leq _{\text{pre}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>pre</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leq _{\text{pre}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50b73be8fc870461f236e6de735bb7611166fa1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.329ex; height:2.676ex;" alt="{\displaystyle \leq _{\text{pre}}}" />". However, just removing the potent induction principle from that axiomatization does not leave a workable algebraic theory. Indeed, even <a href="/wiki/Robinson_arithmetic" title="Robinson arithmetic">Robinson arithmetic</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {Q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">Q</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {Q}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b954e6612a5bc83e108fd4eda3c54901a2786f0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.711ex; height:2.343ex;" alt="{\displaystyle {\mathsf {Q}}}" />, which removes induction but adds back the predecessor existence postulate, does not prove the monoid axiom <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall y.(0+y=y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>y</mi> <mo>.</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>+</mo> <mi>y</mi> <mo>=</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall y.(0+y=y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7df7eee1ef47c93a3f9d5f2281de8571fda94aa2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.704ex; height:2.843ex;" alt="{\displaystyle \forall y.(0+y=y)}" />. <div class="mw-heading mw-heading3"><h3 id="Complete_semirings">Complete semirings</h3>[<a href="/w/index.php?title=Semiring&action=edit&section=15" title="Edit section: Complete semirings">edit</a>]</div> A complete semiring is a semiring for which the additive monoid is a <a href="/wiki/Complete_monoid" class="mw-redirect" title="Complete monoid">complete monoid</a>, meaning that it has an <a href="/wiki/Finitary" title="Finitary">infinitary</a> sum operation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma _{I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma _{I}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac850afe7bf0ebf44ab89440cdab60f9df24d201" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.739ex; height:2.509ex;" alt="{\displaystyle \Sigma _{I}}" /> for any <a href="/wiki/Index_set" title="Index set">index set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" /> and that the following (infinitary) distributive laws must hold:<a href="#cite_note-Kuich11-11">[10]</a><a href="#cite_note-FOOTNOTEDrosteKuich20097–10-12">[11]</a><a href="#cite_note-13">[12]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textstyle \sum }_{i\in I}{\left(a\cdot a_{i}\right)}=a\cdot \left({\textstyle \sum }_{i\in I}{a_{i}}\right),\qquad {\textstyle \sum }_{i\in I}{\left(a_{i}\cdot a\right)}=\left({\textstyle \sum }_{i\in I}{a_{i}}\right)\cdot a.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>∑</mo> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>⋅</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mi>a</mi> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>∑</mo> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="2em"></mspace> <msub> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>∑</mo> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <mi>a</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>∑</mo> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mi>a</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\textstyle \sum }_{i\in I}{\left(a\cdot a_{i}\right)}=a\cdot \left({\textstyle \sum }_{i\in I}{a_{i}}\right),\qquad {\textstyle \sum }_{i\in I}{\left(a_{i}\cdot a\right)}=\left({\textstyle \sum }_{i\in I}{a_{i}}\right)\cdot a.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fccc11e2ea33f29b52d1e05c6bdc0e966cf98102" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:61.255ex; height:3.176ex;" alt="{\displaystyle {\textstyle \sum }_{i\in I}{\left(a\cdot a_{i}\right)}=a\cdot \left({\textstyle \sum }_{i\in I}{a_{i}}\right),\qquad {\textstyle \sum }_{i\in I}{\left(a_{i}\cdot a\right)}=\left({\textstyle \sum }_{i\in I}{a_{i}}\right)\cdot a.}" /></dd></dl> Examples of a complete semiring are the power set of a monoid under union and the matrix semiring over a complete semiring.<a href="#cite_note-FOOTNOTESakarovitch2009471-14">[13]</a> For commutative, additively idempotent and simple semirings, this property is related to <a href="/wiki/Residuated_lattice" title="Residuated lattice">residuated lattices</a>. <div class="mw-heading mw-heading4"><h4 id="Continuous_semirings">Continuous semirings</h4>[<a href="/w/index.php?title=Semiring&action=edit&section=16" title="Edit section: Continuous semirings">edit</a>]</div> A continuous semiring is similarly defined as one for which the addition monoid is a <a href="/wiki/Continuous_monoid" class="mw-redirect" title="Continuous monoid">continuous monoid</a>. That is, partially ordered with the <a href="/wiki/Least-upper-bound_property#Generalization_to_ordered_sets" title="Least-upper-bound property">least upper bound property</a>, and for which addition and multiplication respect order and suprema. The semiring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} \cup \{\infty \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mi mathvariant="normal">∞</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} \cup \{\infty \}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/993fe4a59e08e3c356fef9576ab29e0a499173be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.909ex; height:2.843ex;" alt="{\displaystyle \mathbb {N} \cup \{\infty \}}" /> with usual addition, multiplication and order extended is a continuous semiring.<a href="#cite_note-15">[14]</a> Any continuous semiring is complete:<a href="#cite_note-Kuich11-11">[10]</a> this may be taken as part of the definition.<a href="#cite_note-FOOTNOTESakarovitch2009471-14">[13]</a> <div class="mw-heading mw-heading3"><h3 id="Star_semirings">Star semirings</h3>[<a href="/w/index.php?title=Semiring&action=edit&section=17" title="Edit section: Star semirings">edit</a>]</div> A star semiring (sometimes spelled starsemiring) is a semiring with an additional unary operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/002b545cd6a6f230c91717f3321b3288a405f6a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.054ex; height:2.176ex;" alt="{\displaystyle {}^{*}}" />,<a href="#cite_note-Esik08-10">[9]</a><a href="#cite_note-FOOTNOTEDrosteKuich20097–10-12">[11]</a><a href="#cite_note-16">[15]</a><a href="#cite_note-FOOTNOTEBerstelReutenauer201127-17">[16]</a> satisfying <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{*}=1+aa^{*}=1+a^{*}a.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>a</mi> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>a</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{*}=1+aa^{*}=1+a^{*}a.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83a38a062514780c1150aa71c5b1dda7c4b21b90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:24.161ex; height:2.509ex;" alt="{\displaystyle a^{*}=1+aa^{*}=1+a^{*}a.}" /></dd></dl> A <a href="/wiki/Kleene_algebra" title="Kleene algebra">Kleene algebra</a> is a star semiring with idempotent addition and some additional axioms. They are important in the theory of <a href="/wiki/Formal_language" title="Formal language">formal languages</a> and <a href="/wiki/Regular_expression" title="Regular expression">regular expressions</a>.<a href="#cite_note-FOOTNOTEDrosteKuich20097–10-12">[11]</a> <div class="mw-heading mw-heading4"><h4 id="Complete_star_semirings">Complete star semirings</h4>[<a href="/w/index.php?title=Semiring&action=edit&section=18" title="Edit section: Complete star semirings">edit</a>]</div> In a complete star semiring, the star operator behaves more like the usual <a href="/wiki/Kleene_star" title="Kleene star">Kleene star</a>: for a complete semiring we use the infinitary sum operator to give the usual definition of the Kleene star:<a href="#cite_note-FOOTNOTEDrosteKuich20097–10-12">[11]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{*}={\textstyle \sum }_{j\geq 0}{a^{j}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>∑</mo> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>≥</mo> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{*}={\textstyle \sum }_{j\geq 0}{a^{j}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c72437988cbbd0c4ff66ceacaeefe12f8575b812" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:13.633ex; height:3.676ex;" alt="{\displaystyle a^{*}={\textstyle \sum }_{j\geq 0}{a^{j}},}" /></dd></dl> where <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{j}={\begin{cases}1,&j=0,\\a\cdot a^{j-1}=a^{j-1}\cdot a,&j>0.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>1</mn> <mo>,</mo> </mtd> <mtd> <mi>j</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>a</mi> <mo>⋅</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⋅</mo> <mi>a</mi> <mo>,</mo> </mtd> <mtd> <mi>j</mi> <mo>></mo> <mn>0.</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{j}={\begin{cases}1,&j=0,\\a\cdot a^{j-1}=a^{j-1}\cdot a,&j>0.\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8b686555396f416db3868b0570c3dfe3e83f64a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.964ex; height:6.176ex;" alt="{\displaystyle a^{j}={\begin{cases}1,&j=0,\\a\cdot a^{j-1}=a^{j-1}\cdot a,&j>0.\end{cases}}}" /></dd></dl> Note that star semirings are not related to <a href="/wiki/*-algebra" title="*-algebra">*-algebra</a>, where the star operation should instead be thought of as <a href="/wiki/Complex_conjugation" class="mw-redirect" title="Complex conjugation">complex conjugation</a>. <div class="mw-heading mw-heading4"><h4 id="Conway_semiring">Conway semiring</h4>[<a href="/w/index.php?title=Semiring&action=edit&section=19" title="Edit section: Conway semiring">edit</a>]</div> A Conway semiring is a star semiring satisfying the sum-star and product-star equations:<a href="#cite_note-Esik08-10">[9]</a><a href="#cite_note-18">[17]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}(a+b)^{*}&=\left(a^{*}b\right)^{*}a^{*},\\(ab)^{*}&=1+a(ba)^{*}b.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>b</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>a</mi> <mi>b</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>b</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(a+b)^{*}&=\left(a^{*}b\right)^{*}a^{*},\\(ab)^{*}&=1+a(ba)^{*}b.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ca3091e8fa1c175bab2d2c4f0ffb7cc27fde811" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.217ex; margin-bottom: -0.287ex; width:23.749ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}(a+b)^{*}&=\left(a^{*}b\right)^{*}a^{*},\\(ab)^{*}&=1+a(ba)^{*}b.\end{aligned}}}" /></dd></dl> Every complete star semiring is also a Conway semiring,<a href="#cite_note-FOOTNOTEDrosteKuich200915Theorem_3.4-19">[18]</a> but the converse does not hold. An example of Conway semiring that is not complete is the set of extended non-negative <a href="/wiki/Rational_number" title="Rational number">rational numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{\geq 0}\cup \{\infty \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>≥</mo> <mn>0</mn> </mrow> </msub> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mi mathvariant="normal">∞</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{\geq 0}\cup \{\infty \}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7bc2e0645d90eacd6905550cd4c57b8b811fa15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.372ex; height:3.009ex;" alt="{\displaystyle \mathbb {Q} _{\geq 0}\cup \{\infty \}}" /> with the usual addition and multiplication (this is a modification of the example with extended non-negative reals given in this section by eliminating irrational numbers).<a href="#cite_note-FOOTNOTEDrosteKuich20097–10-12">[11]</a> An iteration semiring is a Conway semiring satisfying the Conway group axioms,<a href="#cite_note-Esik08-10">[9]</a> associated by <a href="/wiki/John_Horton_Conway" title="John Horton Conway">John Conway</a> to groups in star-semirings.<a href="#cite_note-20">[19]</a> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2>[<a href="/w/index.php?title=Semiring&action=edit&section=20" title="Edit section: Examples">edit</a>]</div> <ul><li>By definition, any ring and any semifield is also a semiring.</li> <li>The non-negative elements of a commutative, discretely ordered ring form a commutative, discretely (in the sense defined above) ordered semiring. This includes the non-negative integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }" />.</li> <li>Also the non-negative <a href="/wiki/Rational_number" title="Rational number">rational numbers</a> as well as the non-negative <a href="/wiki/Real_number" title="Real number">real numbers</a> form commutative, ordered semirings.<a href="#cite_note-Gut08-21">[20]</a><a href="#cite_note-FOOTNOTESakarovitch200928-22">[21]</a><a href="#cite_note-FOOTNOTEBerstelReutenauer20114-23">[22]</a> The latter is called <style data-mw-deduplicate="TemplateStyles:r1238216509">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}</style>probability semiring.<a href="#cite_note-FOOTNOTELothaire2005211-7">[6]</a> Neither are rings or distributive lattices. These examples also have multiplicative inverses.</li> <li>New semirings can conditionally be constructed from existing ones, as described. The <a href="/wiki/Extended_natural_numbers" title="Extended natural numbers">extended natural numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} \cup \{\infty \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mi mathvariant="normal">∞</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} \cup \{\infty \}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/993fe4a59e08e3c356fef9576ab29e0a499173be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.909ex; height:2.843ex;" alt="{\displaystyle \mathbb {N} \cup \{\infty \}}" /> with addition and multiplication extended so that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\cdot \infty =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>⋅</mo> <mi mathvariant="normal">∞</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\cdot \infty =0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ca9c473842976ed316eb4b447f69a827d9ed41f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.426ex; height:2.176ex;" alt="{\displaystyle 0\cdot \infty =0}" />.<a href="#cite_note-FOOTNOTESakarovitch200928-22">[21]</a></li> <li>The set of <a href="/wiki/Polynomial" title="Polynomial">polynomials</a> with natural number coefficients, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} [x],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} [x],}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4406ba525ce2c3bc9dd67a1199514d5b9b266391" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.948ex; height:2.843ex;" alt="{\displaystyle \mathbb {N} [x],}" /> forms a commutative semiring. In fact, this is the <a href="/wiki/Free_object" title="Free object">free</a> commutative semiring on a single generator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{x\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15e2fd0e87e3807617e503cce754615838b92215" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.301ex; height:2.843ex;" alt="{\displaystyle \{x\}.}" /> Also polynomials with coefficients in other semirings may be defined, as discussed.</li> <li>The non-negative <a href="/wiki/Positional_notation#Terminating_fractions" title="Positional notation">terminating fractions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\mathbb {N} }{b^{\mathbb {N} }}}:=\left\{mb^{-n}\mid m,n\in \mathbb {N} \right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msup> </mfrac> </mstyle> </mrow> <mo>:=</mo> <mrow> <mo>{</mo> <mrow> <mi>m</mi> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> </mrow> </msup> <mo>∣</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\mathbb {N} }{b^{\mathbb {N} }}}:=\left\{mb^{-n}\mid m,n\in \mathbb {N} \right\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9a333df575ed4b8a6b9b7216ba3c1130e2aa4c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:25.585ex; height:4.176ex;" alt="{\displaystyle {\tfrac {\mathbb {N} }{b^{\mathbb {N} }}}:=\left\{mb^{-n}\mid m,n\in \mathbb {N} \right\}}" />, in a <a href="/wiki/Positional_notation" title="Positional notation">positional number system</a> to a given base <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\in \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07b2dd48f69fc66310b6f437f9f0de49bfb7d787" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.516ex; height:2.176ex;" alt="{\displaystyle b\in \mathbb {N} }" />, form a sub-semiring of the rationals. One has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\mathbb {N} }{b^{\mathbb {N} }}}\subseteq {\tfrac {\mathbb {N} }{c^{\mathbb {N} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msup> </mfrac> </mstyle> </mrow> <mo>⊆</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msup> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\mathbb {N} }{b^{\mathbb {N} }}}\subseteq {\tfrac {\mathbb {N} }{c^{\mathbb {N} }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fd5d9552ed28b7708672e22975f0e6b892ec869" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:8.443ex; height:4.176ex;" alt="{\displaystyle {\tfrac {\mathbb {N} }{b^{\mathbb {N} }}}\subseteq {\tfrac {\mathbb {N} }{c^{\mathbb {N} }}}}" />‍ if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" /> divides <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" />. For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |b|>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |b|>1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87838d1a2091562cfe8bdead8d53039d460518a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.552ex; height:2.843ex;" alt="{\displaystyle |b|>1}" />, the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\mathbb {Z} _{0}}{b^{\mathbb {Z} _{0}}}}:={\tfrac {\mathbb {N} }{b^{\mathbb {N} }}}\cup \left(-{\tfrac {\mathbb {N} _{0}}{b^{\mathbb {N} }}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msup> </mfrac> </mstyle> </mrow> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msup> </mfrac> </mstyle> </mrow> <mo>∪</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msup> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\mathbb {Z} _{0}}{b^{\mathbb {Z} _{0}}}}:={\tfrac {\mathbb {N} }{b^{\mathbb {N} }}}\cup \left(-{\tfrac {\mathbb {N} _{0}}{b^{\mathbb {N} }}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ea831d706613075947fa84849e6dc4a9d0939ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.831ex; height:4.843ex;" alt="{\displaystyle {\tfrac {\mathbb {Z} _{0}}{b^{\mathbb {Z} _{0}}}}:={\tfrac {\mathbb {N} }{b^{\mathbb {N} }}}\cup \left(-{\tfrac {\mathbb {N} _{0}}{b^{\mathbb {N} }}}\right)}" /> is the ring of all terminating fractions to base <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdb96677ba71b937617ca8751955f884f6306b64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.644ex; height:2.509ex;" alt="{\displaystyle b,}" /> and is <a href="/wiki/Dense_set" title="Dense set">dense</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" />.</li> <li>The <a href="/wiki/Log_semiring" title="Log semiring">log semiring</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} \cup \{\pm \infty \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mo>±</mo> <mi mathvariant="normal">∞</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} \cup \{\pm \infty \}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f2ca1c8ea51b89aba6b5c8244ffd1f9d50410d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.717ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} \cup \{\pm \infty \}}" /> with addition given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\oplus y=-\log \left(e^{-x}+e^{-y}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>⊕</mo> <mi>y</mi> <mo>=</mo> <mo>−</mo> <mi>log</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>x</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>y</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\oplus y=-\log \left(e^{-x}+e^{-y}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e652f17b7e06c5df195ae5af0f0b20ef4c8ec1fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.507ex; height:3.176ex;" alt="{\displaystyle x\oplus y=-\log \left(e^{-x}+e^{-y}\right)}" /> with multiplication <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e863acd450b409ef6564ff90998f5371e205731e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.455ex; height:2.343ex;" alt="{\displaystyle +,}" /> zero element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +\infty ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +\infty ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0aba80b6bc968c1bbc0df9f6c4f70b63ad261317" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.779ex; height:2.343ex;" alt="{\displaystyle +\infty ,}" /> and unit element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/916e773e0593223c306a3e6852348177d1934962" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.809ex; height:2.176ex;" alt="{\displaystyle 0.}" /><a href="#cite_note-FOOTNOTELothaire2005211-7">[6]</a></li></ul> Similarly, in the presence of an appropriate order with bottom element, <ul><li><a href="/wiki/Tropical_semiring" title="Tropical semiring">Tropical semirings</a> are variously defined. The max-plus semiring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} \cup \{-\infty \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} \cup \{-\infty \}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d803f13229b881b5f52a24255a3a74f0368af546" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.717ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} \cup \{-\infty \}}" /> is a commutative semiring with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \max(a,b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \max(a,b)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0603f0969a6eb143615dc0339002987aa0dd9daa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.396ex; height:2.843ex;" alt="{\displaystyle \max(a,b)}" /> serving as semiring addition (identity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle -\infty }" />) and ordinary addition (identity 0) serving as semiring multiplication. In an alternative formulation, the tropical semiring is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} \cup \{\infty \},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mi mathvariant="normal">∞</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} \cup \{\infty \},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2359b75db1364467e2f29487b26e158506426d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.556ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} \cup \{\infty \},}" /> and min replaces max as the addition operation.<a href="#cite_note-24">[23]</a> A related version has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} \cup \{\pm \infty \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mo>±</mo> <mi mathvariant="normal">∞</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} \cup \{\pm \infty \}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f2ca1c8ea51b89aba6b5c8244ffd1f9d50410d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.717ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} \cup \{\pm \infty \}}" /> as the underlying set.<a href="#cite_note-FOOTNOTELothaire2005211-7">[6]</a><a href="#cite_note-Kuich11-11">[10]</a> They are an active area of research, linking <a href="/wiki/Algebraic_variety" title="Algebraic variety">algebraic varieties</a> with <a href="/wiki/Piecewise_linear_manifold" title="Piecewise linear manifold">piecewise linear</a> structures.<a href="#cite_note-25">[24]</a></li> <li>The <a href="/wiki/Jan_%C5%81ukasiewicz" title="Jan Łukasiewicz">Łukasiewicz</a> semiring: the closed interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,1]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/738f7d23bb2d9642bab520020873cccbef49768d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle [0,1]}" /> with addition of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" /> given by taking the maximum of the arguments (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \max(a,b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \max(a,b)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0603f0969a6eb143615dc0339002987aa0dd9daa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.396ex; height:2.843ex;" alt="{\displaystyle \max(a,b)}" />) and multiplication of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" /> given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \max(0,a+b-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \max(0,a+b-1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d09149841f6c0e3e3acbadb6eca51dccb13fd935" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.402ex; height:2.843ex;" alt="{\displaystyle \max(0,a+b-1)}" /> appears in <a href="/wiki/Multi-valued_logic" class="mw-redirect" title="Multi-valued logic">multi-valued logic</a>.<a href="#cite_note-FOOTNOTEDrosteKuich20097–10-12">[11]</a></li> <li>The <a href="/wiki/Andrew_Viterbi" title="Andrew Viterbi">Viterbi</a> semiring is also defined over the base set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,1]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/738f7d23bb2d9642bab520020873cccbef49768d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle [0,1]}" /> and has the maximum as its addition, but its multiplication is the usual multiplication of real numbers. It appears in <a href="/wiki/Probabilistic_parsing" class="mw-redirect" title="Probabilistic parsing">probabilistic parsing</a>.<a href="#cite_note-FOOTNOTEDrosteKuich20097–10-12">[11]</a></li></ul> <ul><li>The set of all ideals of a given semiring form a semiring under addition and multiplication of ideals.</li> <li>Any bounded, distributive lattice is a commutative, semiring under join and meet. A Boolean algebra is a special case of these. A <a href="/wiki/Boolean_ring" title="Boolean ring">Boolean ring</a> is also a semiring (indeed, a ring) but it is not idempotent under addition. A Boolean semiring is a semiring isomorphic to a sub-semiring of a Boolean algebra.<a href="#cite_note-Gut08-21">[20]</a></li> <li>The commutative semiring formed by the two-element Boolean algebra and defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+1=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+1=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5120dbef0aa4b1760be59a33516661780dfc7a14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.426ex; height:2.343ex;" alt="{\displaystyle 1+1=1}" />. It is also called the <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509" />Boolean semiring.<a href="#cite_note-FOOTNOTELothaire2005211-7">[6]</a><a href="#cite_note-FOOTNOTESakarovitch200928-22">[21]</a><a href="#cite_note-FOOTNOTEBerstelReutenauer20114-23">[22]</a><a href="#cite_note-Esik08-10">[9]</a> Now given two sets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3765557b7effa1a5f2f4dce9c80a25973b7009f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.42ex; height:2.509ex;" alt="{\displaystyle Y,}" /> <a href="/wiki/Binary_relation" title="Binary relation">binary relations</a> between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" /> correspond to matrices indexed by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" /> with entries in the Boolean semiring, <a href="/wiki/Matrix_addition" title="Matrix addition">matrix addition</a> corresponds to union of relations, and <a href="/wiki/Matrix_multiplication" title="Matrix multiplication">matrix multiplication</a> corresponds to <a href="/wiki/Composition_of_relations" title="Composition of relations">composition of relations</a>.<a href="#cite_note-26">[25]</a></li> <li>Any <a href="/wiki/Quantale" title="Quantale">unital quantale</a> is a semiring under join and multiplication.</li> <li>A normal <a href="/wiki/Skew_lattice" title="Skew lattice">skew lattice</a> in a ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" /> is a semiring for the operations multiplication and nabla, where the latter operation is defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\nabla b=a+b+ba-aba-bab}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi mathvariant="normal">∇</mi> <mi>b</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>+</mo> <mi>b</mi> <mi>a</mi> <mo>−</mo> <mi>a</mi> <mi>b</mi> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mi>a</mi> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\nabla b=a+b+ba-aba-bab}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dedb587ba55eac28466037c092a5993b4d7089e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:29.76ex; height:2.343ex;" alt="{\displaystyle a\nabla b=a+b+ba-aba-bab}" /></li></ul> More using monoids, <ul><li>The construction of semirings <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {End} (M)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>End</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {End} (M)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a654fdfacc063d9cb412bfac7b28eb2198b6f909" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.419ex; height:2.843ex;" alt="{\displaystyle \operatorname {End} (M)}" /> from a commutative monoid <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" /> has been described. As noted, give a semiring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" />, the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}" /> matrices form another semiring. For example, the matrices with non-negative entries, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}}_{n}(\mathbb {N} ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}_{n}(\mathbb {N} ),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/684b46042783b874fc0b21f75ea3e4a2d10fec1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.143ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M}}_{n}(\mathbb {N} ),}" /> form a matrix semiring.<a href="#cite_note-Gut08-21">[20]</a></li> <li>Given an alphabet (finite set) Σ, the set of <a href="/wiki/Formal_language" title="Formal language">formal languages</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e1f558f53cda207614abdf90162266c70bc5c1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }" /> (subsets of <a href="/wiki/Kleene_star" title="Kleene star"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma ^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma ^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/807344600a40f1de7136f8b54576e12e9428bef4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.343ex;" alt="{\displaystyle \Sigma ^{*}}" /></a>) is a semiring with product induced by <a href="/wiki/String_concatenation" class="mw-redirect" title="String concatenation">string concatenation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{1}\cdot L_{2}=\left\{w_{1}w_{2}\mid w_{1}\in L_{1},w_{2}\in L_{2}\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∣</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∈</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∈</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{1}\cdot L_{2}=\left\{w_{1}w_{2}\mid w_{1}\in L_{1},w_{2}\in L_{2}\right\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/854d0a7caddba4133d8040c2182cf058acf31d50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.177ex; height:2.843ex;" alt="{\displaystyle L_{1}\cdot L_{2}=\left\{w_{1}w_{2}\mid w_{1}\in L_{1},w_{2}\in L_{2}\right\}}" /> and addition as the union of languages (that is, ordinary union as sets). The zero of this semiring is the empty set (empty language) and the semiring's unit is the language containing only the <a href="/wiki/Empty_string" title="Empty string">empty string</a>.<a href="#cite_note-FOOTNOTEDrosteKuich20097–10-12">[11]</a></li> <li>Generalizing the previous example (by viewing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma ^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma ^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/807344600a40f1de7136f8b54576e12e9428bef4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.343ex;" alt="{\displaystyle \Sigma ^{*}}" /> as the <a href="/wiki/Free_monoid" title="Free monoid">free monoid</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e1f558f53cda207614abdf90162266c70bc5c1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }" />), take <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" /> to be any monoid; the power set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wp (M)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">℘</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wp (M)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c785d5508afb82268b23b40d2025798bd1d64e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.73ex; height:2.843ex;" alt="{\displaystyle \wp (M)}" /> of all subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" /> forms a semiring under set-theoretic union as addition and set-wise multiplication: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\cdot V=\{u\cdot v\mid u\in U,\ v\in V\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>⋅</mo> <mi>V</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>u</mi> <mo>⋅</mo> <mi>v</mi> <mo>∣</mo> <mi>u</mi> <mo>∈</mo> <mi>U</mi> <mo>,</mo> <mtext> </mtext> <mi>v</mi> <mo>∈</mo> <mi>V</mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\cdot V=\{u\cdot v\mid u\in U,\ v\in V\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4badebc525b4dc53d774a713536b95b70836527" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.716ex; height:2.843ex;" alt="{\displaystyle U\cdot V=\{u\cdot v\mid u\in U,\ v\in V\}.}" /><a href="#cite_note-FOOTNOTEBerstelReutenauer20114-23">[22]</a></li> <li>Similarly, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (M,e,\cdot )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>M</mi> <mo>,</mo> <mi>e</mi> <mo>,</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (M,e,\cdot )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbf4091e195cccfa19d6730f28f1ea4d4c9862d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.05ex; height:2.843ex;" alt="{\displaystyle (M,e,\cdot )}" /> is a monoid, then the set of finite <a href="/wiki/Multiset" title="Multiset">multisets</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" /> forms a semiring. That is, an element is a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\mid M\to \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∣</mo> <mi>M</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\mid M\to \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e94491e3a7faadb334bbe7eac28ab52614c15b8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.95ex; height:2.843ex;" alt="{\displaystyle f\mid M\to \mathbb {N} }" />; given an element of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b466e90209f39c0c2caad1b11445824b82c2f536" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.089ex; height:2.509ex;" alt="{\displaystyle M,}" /> the function tells you how many times that element occurs in the multiset it represents. The additive unit is the constant zero function. The multiplicative unit is the function mapping <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" /> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cc5fd8163a83100c5330622e9e317fa4e872403" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.809ex; height:2.509ex;" alt="{\displaystyle 1,}" /> and all other elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" /> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/916e773e0593223c306a3e6852348177d1934962" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.809ex; height:2.176ex;" alt="{\displaystyle 0.}" /> The sum is given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f+g)(x)=f(x)+g(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>+</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f+g)(x)=f(x)+g(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf80cb50218eac1e40d4a0908bd039db3bd0863c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.795ex; height:2.843ex;" alt="{\displaystyle (f+g)(x)=f(x)+g(x)}" /> and the product is given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (fg)(x)=\sum \{f(y)g(z)\mid y\cdot z=x\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>∑</mo> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>∣</mo> <mi>y</mi> <mo>⋅</mo> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (fg)(x)=\sum \{f(y)g(z)\mid y\cdot z=x\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f677c658b5b176e3029dbf40a90d6183057895eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:35.313ex; height:3.843ex;" alt="{\displaystyle (fg)(x)=\sum \{f(y)g(z)\mid y\cdot z=x\}.}" /></li></ul> Regarding sets and similar abstractions, <ul><li>Given a set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59c21c569e498e0197798e3428b2bc6f25c0a457" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.429ex; height:2.509ex;" alt="{\displaystyle U,}" /> the set of <a href="/wiki/Binary_relation" title="Binary relation">binary relations</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" /> is a semiring with addition the union (of relations as sets) and multiplication the <a href="/wiki/Composition_of_relations" title="Composition of relations">composition of relations</a>. The semiring's zero is the <a href="/wiki/Empty_relation" class="mw-redirect" title="Empty relation">empty relation</a> and its unit is the <a href="/wiki/Identity_relation" class="mw-redirect" title="Identity relation">identity relation</a>.<a href="#cite_note-FOOTNOTEDrosteKuich20097–10-12">[11]</a> These relations correspond to the <a href="/wiki/Matrix_semiring" class="mw-redirect" title="Matrix semiring">matrix semiring</a> (indeed, matrix semialgebra) of <a href="/wiki/Square_matrices" class="mw-redirect" title="Square matrices">square matrices</a> indexed by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" /> with entries in the Boolean semiring, and then addition and multiplication are the usual matrix operations, while zero and the unit are the usual <a href="/wiki/Zero_matrix" title="Zero matrix">zero matrix</a> and <a href="/wiki/Identity_matrix" title="Identity matrix">identity matrix</a>.</li> <li>The set of <a href="/wiki/Cardinal_number" title="Cardinal number">cardinal numbers</a> smaller than any given <a href="/wiki/Infinity" title="Infinity">infinite</a> cardinal form a semiring under cardinal addition and multiplication. The class of all cardinals of an <a href="/wiki/Inner_model" title="Inner model">inner model</a> form a (class) semiring under (inner model) cardinal addition and multiplication.</li> <li>The family of (isomorphism equivalence classes of) <a href="/wiki/Combinatorial_class" title="Combinatorial class">combinatorial classes</a> (sets of countably many objects with non-negative integer sizes such that there are finitely many objects of each size) with the empty class as the zero object, the class consisting only of the empty set as the unit, <a href="/wiki/Disjoint_union" title="Disjoint union">disjoint union</a> of classes as addition, and <a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a> of classes as multiplication.<a href="#cite_note-27">[26]</a></li> <li>Isomorphism classes of objects in any <a href="/wiki/Distributive_category" title="Distributive category">distributive category</a>, under <a href="/wiki/Coproduct" title="Coproduct">coproduct</a> and <a href="/wiki/Product_(category_theory)" title="Product (category theory)">product</a> operations, form a semiring known as a Burnside rig.<a href="#cite_note-28">[27]</a> A Burnside rig is a ring if and only if the category is <a href="/wiki/Category_of_small_categories" title="Category of small categories">trivial</a>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Star_semirings_2">Star semirings</h3>[<a href="/w/index.php?title=Semiring&action=edit&section=21" title="Edit section: Star semirings">edit</a>]</div> Several structures mentioned above can be equipped with a star operation. <ul><li>The <a href="#binary_relations">aforementioned</a> semiring of <a href="/wiki/Binary_relation" title="Binary relation">binary relations</a> over some base set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" /> in which <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{*}=\bigcup _{n\geq 0}R^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <munder> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>≥</mo> <mn>0</mn> </mrow> </munder> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{*}=\bigcup _{n\geq 0}R^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d25602fb5bc862e5af406af75e7b3a64652933c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:12.373ex; height:5.676ex;" alt="{\displaystyle R^{*}=\bigcup _{n\geq 0}R^{n}}" /> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\subseteq U\times U.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>⊆</mo> <mi>U</mi> <mo>×</mo> <mi>U</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\subseteq U\times U.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8eb760b02c58167d1db016d887e6a9b38e64608a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.915ex; height:2.343ex;" alt="{\displaystyle R\subseteq U\times U.}" /> This star operation is actually the <a href="/wiki/Reflexive_closure" title="Reflexive closure">reflexive</a> and <a href="/wiki/Transitive_closure" title="Transitive closure">transitive closure</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" /> (that is, the smallest reflexive and transitive binary relation over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" /> containing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdcae6b33a27f86c7961318cd7ee3d789d3bcdd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.411ex; height:2.176ex;" alt="{\displaystyle R.}" />).<a href="#cite_note-FOOTNOTEDrosteKuich20097–10-12">[11]</a></li> <li>The <a href="#formal_languages">semiring of formal languages</a> is also a complete star semiring, with the star operation coinciding with the Kleene star (for sets/languages).<a href="#cite_note-FOOTNOTEDrosteKuich20097–10-12">[11]</a></li> <li>The set of non-negative <a href="/wiki/Extended_real" class="mw-redirect" title="Extended real">extended reals</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,\infty ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,\infty ]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52088d5605716e18068a460dec118214954a68e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.814ex; height:2.843ex;" alt="{\displaystyle [0,\infty ]}" /> together with the usual addition and multiplication of reals is a complete star semiring with the star operation given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{*}={\tfrac {1}{1-a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mi>a</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{*}={\tfrac {1}{1-a}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2299b2a398185378ea0af3f0aa3ce4718400e40e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:9.189ex; height:3.676ex;" alt="{\displaystyle a^{*}={\tfrac {1}{1-a}}}" /> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq a<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>a</mi> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq a<1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab3c255b7ccb797bc700db9fc7110a5f78256308" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.752ex; height:2.343ex;" alt="{\displaystyle 0\leq a<1}" /> (that is, the <a href="/wiki/Geometric_series" title="Geometric series">geometric series</a>) and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{*}=\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{*}=\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12f155028942545e2995c06275f595a713045ac4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.706ex; height:2.343ex;" alt="{\displaystyle a^{*}=\infty }" /> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\geq 1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≥</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\geq 1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/299ff497b970b982a5a81e099b0ceae0749fa5fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.138ex; height:2.343ex;" alt="{\displaystyle a\geq 1.}" /><a href="#cite_note-FOOTNOTEDrosteKuich20097–10-12">[11]</a></li> <li>The Boolean semiring with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0^{*}=1^{*}=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0^{*}=1^{*}=1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/203bcab623708f6ee5207aa36dcdc07519f04892" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.44ex; height:2.343ex;" alt="{\displaystyle 0^{*}=1^{*}=1.}" /><a href="#cite_note-conway-29">[b]</a><a href="#cite_note-FOOTNOTEDrosteKuich20097–10-12">[11]</a></li> <li>The semiring on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} \cup \{\infty \},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mi mathvariant="normal">∞</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} \cup \{\infty \},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2d6c148645f2ab419218ae9edd5599d05b45861" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.556ex; height:2.843ex;" alt="{\displaystyle \mathbb {N} \cup \{\infty \},}" /> with extended addition and multiplication, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0^{*}=1,a^{*}=\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0^{*}=1,a^{*}=\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dbc8f8bbacb6105db4e149f95fbe91de9254541" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.218ex; height:2.676ex;" alt="{\displaystyle 0^{*}=1,a^{*}=\infty }" /> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\geq 1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≥</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\geq 1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/299ff497b970b982a5a81e099b0ceae0749fa5fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.138ex; height:2.343ex;" alt="{\displaystyle a\geq 1.}" /><a href="#cite_note-conway-29">[b]</a><a href="#cite_note-FOOTNOTEDrosteKuich20097–10-12">[11]</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2>[<a href="/w/index.php?title=Semiring&action=edit&section=22" title="Edit section: Applications">edit</a>]</div> The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\max ,+)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo movablelimits="true" form="prefix">max</mo> <mo>,</mo> <mo>+</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\max ,+)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9e42d91e5020b82a45c62ae5558a4b5c433a158" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.977ex; height:2.843ex;" alt="{\displaystyle (\max ,+)}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\min ,+)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo movablelimits="true" form="prefix">min</mo> <mo>,</mo> <mo>+</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\min ,+)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f278b2987c1d32a558232b1ea34db6345c4e659" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.527ex; height:2.843ex;" alt="{\displaystyle (\min ,+)}" /> <a href="/wiki/Tropical_semiring" title="Tropical semiring">tropical semirings</a> on the reals are often used in <a href="/wiki/Performance_evaluation" class="mw-redirect" title="Performance evaluation">performance evaluation</a> on discrete event systems. The real numbers then are the "costs" or "arrival time"; the "max" operation corresponds to having to wait for all prerequisites of an events (thus taking the maximal time) while the "min" operation corresponds to being able to choose the best, less costly choice; and + corresponds to accumulation along the same path. The <a href="/wiki/Floyd%E2%80%93Warshall_algorithm" title="Floyd–Warshall algorithm">Floyd–Warshall algorithm</a> for <a href="/wiki/Shortest_path" class="mw-redirect" title="Shortest path">shortest paths</a> can thus be reformulated as a computation over a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\min ,+)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo movablelimits="true" form="prefix">min</mo> <mo>,</mo> <mo>+</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\min ,+)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f278b2987c1d32a558232b1ea34db6345c4e659" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.527ex; height:2.843ex;" alt="{\displaystyle (\min ,+)}" /> algebra. Similarly, the <a href="/wiki/Viterbi_algorithm" title="Viterbi algorithm">Viterbi algorithm</a> for finding the most probable state sequence corresponding to an observation sequence in a <a href="/wiki/Hidden_Markov_model" title="Hidden Markov model">hidden Markov model</a> can also be formulated as a computation over a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\max ,\times )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo movablelimits="true" form="prefix">max</mo> <mo>,</mo> <mo>×</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\max ,\times )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e73860e2e4eadc96dbf7278d7fdcd08c3a25f8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.977ex; height:2.843ex;" alt="{\displaystyle (\max ,\times )}" /> algebra on probabilities. These <a href="/wiki/Dynamic_programming" title="Dynamic programming">dynamic programming</a> algorithms rely on the <a href="/wiki/Distributive_property" title="Distributive property">distributive property</a> of their associated semirings to compute quantities over a large (possibly exponential) number of terms more efficiently than enumerating each of them.<a href="#cite_note-FOOTNOTEPair1967271-30">[28]</a><a href="#cite_note-FOOTNOTEDerniamePair1971-31">[29]</a> <div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2>[<a href="/w/index.php?title=Semiring&action=edit&section=23" title="Edit section: Generalizations">edit</a>]</div> A generalization of semirings does not require the existence of a multiplicative identity, so that multiplication is a <a href="/wiki/Semigroup" title="Semigroup">semigroup</a> rather than a monoid. Such structures are called hemirings<a href="#cite_note-FOOTNOTEGolan19991Ch_1-32">[30]</a> or pre-semirings.<a href="#cite_note-FOOTNOTEGondranMinoux200822Ch_1,_§4.2-33">[31]</a> A further generalization are left-pre-semirings,<a href="#cite_note-FOOTNOTEGondranMinoux200820Ch_1,_§4.1-34">[32]</a> which additionally do not require right-distributivity (or right-pre-semirings, which do not require left-distributivity). Yet a further generalization are <a href="/wiki/Near-semiring" title="Near-semiring">near-semirings</a>: in addition to not requiring a neutral element for product, or right-distributivity (or left-distributivity), they do not require addition to be commutative. Just as cardinal numbers form a (class) semiring, so do <a href="/wiki/Ordinal_number" title="Ordinal number">ordinal numbers</a> form a <a href="/wiki/Near-semiring" title="Near-semiring">near-semiring</a>, when the standard <a href="/wiki/Ordinal_arithmetic" title="Ordinal arithmetic">ordinal addition and multiplication</a> are taken into account. However, the class of ordinals can be turned into a semiring by considering the so-called <a href="/wiki/Ordinal_arithmetic#Natural_operations" title="Ordinal arithmetic">natural (or Hessenberg) operations</a> instead. In <a href="/wiki/Category_theory" title="Category theory">category theory</a>, a 2-rig is a category with <a href="/wiki/Functor" title="Functor">functorial</a> operations analogous to those of a rig. That the cardinal numbers form a rig can be categorified to say that the <a href="/wiki/Category_of_sets" title="Category of sets">category of sets</a> (or more generally, any <a href="/wiki/Topos" title="Topos">topos</a>) is a 2-rig. <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Semiring&action=edit&section=24" title="Edit section: See also">edit</a>]</div> <ul><li><a href="/wiki/Ring_of_sets" title="Ring of sets">Ring of sets</a> – Family closed under unions and relative complements</li> <li><a href="/wiki/Valuation_algebra" class="mw-redirect" title="Valuation algebra">Valuation algebra</a> – Algebra describing information processingPages displaying short descriptions of redirect targets</li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Semiring&action=edit&section=25" title="Edit section: Notes">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-2"><a href="#cite_ref-2">^</a> <a rel="nofollow" class="external text" href="http://www.proofwiki.org/wiki/Definition:Rig">For an example see the definition of rig on Proofwiki.org</a> </li> <li id="cite_note-conway-29">^ <a href="#cite_ref-conway_29-0">a</a> <a href="#cite_ref-conway_29-1">b</a> This is a complete star semiring and thus also a Conway semiring.<a href="#cite_note-FOOTNOTEDrosteKuich20097–10-12">[11]</a> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Citations">Citations</h2>[<a href="/w/index.php?title=Semiring&action=edit&section=26" title="Edit section: Citations">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626" /><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-FOOTNOTEGłazek20027-1"><a href="#cite_ref-FOOTNOTEGłazek20027_1-0">^</a> <a href="#CITEREFGłazek2002">Głazek (2002)</a>, p. 7 </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFKuntzmann1972" class="citation book cs1 cs1-prop-foreign-lang-source">Kuntzmann, J. (1972). Théorie des réseaux (graphes) (in French). Paris: Dunod. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0239.05101">0239.05101</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Th%C3%A9orie+des+r%C3%A9seaux+%28graphes%29&rft.place=Paris&rft.pub=Dunod&rft.date=1972&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0239.05101%23id-name%3DZbl&rft.aulast=Kuntzmann&rft.aufirst=J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemiring" class="Z3988"> </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> <a rel="nofollow" class="external text" href="http://marcpouly.ch/pdf/internal_100712.pdf">Semirings for breakfast</a>, slide 17 </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBaccelliOlsderQuadratCohen1992" class="citation book cs1">Baccelli, François Louis; Olsder, Geert Jan; Quadrat, Jean-Pierre; Cohen, Guy (1992). Synchronization and linearity. An algebra for discrete event systems. Wiley Series on Probability and Mathematical Statistics. Chichester: Wiley. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0824.93003">0824.93003</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Synchronization+and+linearity.+An+algebra+for+discrete+event+systems&rft.place=Chichester&rft.series=Wiley+Series+on+Probability+and+Mathematical+Statistics&rft.pub=Wiley&rft.date=1992&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0824.93003%23id-name%3DZbl&rft.aulast=Baccelli&rft.aufirst=Fran%C3%A7ois+Louis&rft.au=Olsder%2C+Geert+Jan&rft.au=Quadrat%2C+Jean-Pierre&rft.au=Cohen%2C+Guy&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemiring" class="Z3988"> </li> <li id="cite_note-FOOTNOTEBerstelPerrin1985[httpsbooksgooglecombooksidGHJHqezwwpcCpgPA26dq22asemiringKisasetequippedwithtwooperations22_p._26]-6"><a href="#cite_ref-FOOTNOTEBerstelPerrin1985[httpsbooksgooglecombooksidGHJHqezwwpcCpgPA26dq22asemiringKisasetequippedwithtwooperations22_p._26]_6-0">^</a> <a href="#CITEREFBerstelPerrin1985">Berstel & Perrin (1985)</a>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=GHJHqezwwpcC&pg=PA26&dq=%22a+semiring+K+is+a+set+equipped+with+two+operations%22">p. 26</a> </li> <li id="cite_note-FOOTNOTELothaire2005211-7">^ <a href="#cite_ref-FOOTNOTELothaire2005211_7-0">a</a> <a href="#cite_ref-FOOTNOTELothaire2005211_7-1">b</a> <a href="#cite_ref-FOOTNOTELothaire2005211_7-2">c</a> <a href="#cite_ref-FOOTNOTELothaire2005211_7-3">d</a> <a href="#cite_ref-FOOTNOTELothaire2005211_7-4">e</a> <a href="#CITEREFLothaire2005">Lothaire (2005)</a>, p. 211 </li> <li id="cite_note-FOOTNOTESakarovitch200927–28-8"><a href="#cite_ref-FOOTNOTESakarovitch200927–28_8-0">^</a> <a href="#CITEREFSakarovitch2009">Sakarovitch (2009)</a>, pp. 27–28 </li> <li id="cite_note-FOOTNOTELothaire2005212-9"><a href="#cite_ref-FOOTNOTELothaire2005212_9-0">^</a> <a href="#CITEREFLothaire2005">Lothaire (2005)</a>, p. 212 </li> <li id="cite_note-Esik08-10">^ <a href="#cite_ref-Esik08_10-0">a</a> <a href="#cite_ref-Esik08_10-1">b</a> <a href="#cite_ref-Esik08_10-2">c</a> <a href="#cite_ref-Esik08_10-3">d</a> <a href="#cite_ref-Esik08_10-4">e</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFÉsik2008" class="citation book cs1">Ésik, Zoltán (2008). "Iteration semirings". In Ito, Masami (ed.). Developments in language theory. 12th international conference, DLT 2008, Kyoto, Japan, September 16–19, 2008. Proceedings. Lecture Notes in Computer Science. Vol. 5257. Berlin: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. pp. 1–20. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-540-85780-8_1">10.1007/978-3-540-85780-8_1</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-85779-2" title="Special:BookSources/978-3-540-85779-2"><bdi>978-3-540-85779-2</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1161.68598">1161.68598</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Iteration+semirings&rft.btitle=Developments+in+language+theory.+12th+international+conference%2C+DLT+2008%2C+Kyoto%2C+Japan%2C+September+16%E2%80%9319%2C+2008.+Proceedings&rft.place=Berlin&rft.series=Lecture+Notes+in+Computer+Science&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1-%3C%2Fspan%3E20&rft.pub=Springer-Verlag&rft.date=2008&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1161.68598%23id-name%3DZbl&rft_id=info%3Adoi%2F10.1007%2F978-3-540-85780-8_1&rft.isbn=978-3-540-85779-2&rft.aulast=%C3%89sik&rft.aufirst=Zolt%C3%A1n&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemiring" class="Z3988"> </li> <li id="cite_note-Kuich11-11">^ <a href="#cite_ref-Kuich11_11-0">a</a> <a href="#cite_ref-Kuich11_11-1">b</a> <a href="#cite_ref-Kuich11_11-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKuich2011" class="citation book cs1">Kuich, Werner (2011). "Algebraic systems and pushdown automata". In Kuich, Werner (ed.). Algebraic foundations in computer science. Essays dedicated to Symeon Bozapalidis on the occasion of his retirement. Lecture Notes in Computer Science. Vol. 7020. Berlin: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. pp. 228–256. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-24896-2" title="Special:BookSources/978-3-642-24896-2"><bdi>978-3-642-24896-2</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1251.68135">1251.68135</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Algebraic+systems+and+pushdown+automata&rft.btitle=Algebraic+foundations+in+computer+science.+Essays+dedicated+to+Symeon+Bozapalidis+on+the+occasion+of+his+retirement&rft.place=Berlin&rft.series=Lecture+Notes+in+Computer+Science&rft.pages=%3Cspan+class%3D%22nowrap%22%3E228-%3C%2Fspan%3E256&rft.pub=Springer-Verlag&rft.date=2011&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1251.68135%23id-name%3DZbl&rft.isbn=978-3-642-24896-2&rft.aulast=Kuich&rft.aufirst=Werner&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemiring" class="Z3988"> </li> <li id="cite_note-FOOTNOTEDrosteKuich20097–10-12">^ <a href="#cite_ref-FOOTNOTEDrosteKuich20097–10_12-0">a</a> <a href="#cite_ref-FOOTNOTEDrosteKuich20097–10_12-1">b</a> <a href="#cite_ref-FOOTNOTEDrosteKuich20097–10_12-2">c</a> <a href="#cite_ref-FOOTNOTEDrosteKuich20097–10_12-3">d</a> <a href="#cite_ref-FOOTNOTEDrosteKuich20097–10_12-4">e</a> <a href="#cite_ref-FOOTNOTEDrosteKuich20097–10_12-5">f</a> <a href="#cite_ref-FOOTNOTEDrosteKuich20097–10_12-6">g</a> <a href="#cite_ref-FOOTNOTEDrosteKuich20097–10_12-7">h</a> <a href="#cite_ref-FOOTNOTEDrosteKuich20097–10_12-8">i</a> <a href="#cite_ref-FOOTNOTEDrosteKuich20097–10_12-9">j</a> <a href="#cite_ref-FOOTNOTEDrosteKuich20097–10_12-10">k</a> <a href="#cite_ref-FOOTNOTEDrosteKuich20097–10_12-11">l</a> <a href="#cite_ref-FOOTNOTEDrosteKuich20097–10_12-12">m</a> <a href="#cite_ref-FOOTNOTEDrosteKuich20097–10_12-13">n</a> <a href="#cite_ref-FOOTNOTEDrosteKuich20097–10_12-14">o</a> <a href="#CITEREFDrosteKuich2009">Droste & Kuich (2009)</a>, pp. 7–10 </li> <li id="cite_note-13"><a href="#cite_ref-13">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKuich1990" class="citation book cs1">Kuich, Werner (1990). <a rel="nofollow" class="external text" href="https://archive.org/details/automatalanguage0000ical/page/103">"ω-continuous semirings, algebraic systems and pushdown automata"</a>. In Paterson, Michael S. (ed.). Automata, Languages and Programming: 17th International Colloquium, Warwick University, England, July 16–20, 1990, Proceedings. Lecture Notes in Computer Science. Vol. 443. <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/automatalanguage0000ical/page/103">103–110</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-52826-1" title="Special:BookSources/3-540-52826-1"><bdi>3-540-52826-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=%CF%89-continuous+semirings%2C+algebraic+systems+and+pushdown+automata&rft.btitle=Automata%2C+Languages+and+Programming%3A+17th+International+Colloquium%2C+Warwick+University%2C+England%2C+July+16%E2%80%9320%2C+1990%2C+Proceedings&rft.series=Lecture+Notes+in+Computer+Science&rft.pages=103-110&rft.pub=Springer-Verlag&rft.date=1990&rft.isbn=3-540-52826-1&rft.aulast=Kuich&rft.aufirst=Werner&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fautomatalanguage0000ical%2Fpage%2F103&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemiring" class="Z3988"> </li> <li id="cite_note-FOOTNOTESakarovitch2009471-14">^ <a href="#cite_ref-FOOTNOTESakarovitch2009471_14-0">a</a> <a href="#cite_ref-FOOTNOTESakarovitch2009471_14-1">b</a> <a href="#CITEREFSakarovitch2009">Sakarovitch (2009)</a>, p. 471 </li> <li id="cite_note-15"><a href="#cite_ref-15">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFÉsikLeiß2002" class="citation book cs1">Ésik, Zoltán; Leiß, Hans (2002). "Greibach normal form in algebraically complete semirings". In Bradfield, Julian (ed.). Computer science logic. 16th international workshop, CSL 2002, 11th annual conference of the EACSL, Edinburgh, Scotland, September 22–25, 2002. Proceedings. Lecture Notes in Computer Science. Vol. 2471. Berlin: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. pp. 135–150. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1020.68056">1020.68056</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Greibach+normal+form+in+algebraically+complete+semirings&rft.btitle=Computer+science+logic.+16th+international+workshop%2C+CSL+2002%2C+11th+annual+conference+of+the+EACSL%2C+Edinburgh%2C+Scotland%2C+September+22%E2%80%9325%2C+2002.+Proceedings&rft.place=Berlin&rft.series=Lecture+Notes+in+Computer+Science&rft.pages=%3Cspan+class%3D%22nowrap%22%3E135-%3C%2Fspan%3E150&rft.pub=Springer-Verlag&rft.date=2002&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1020.68056%23id-name%3DZbl&rft.aulast=%C3%89sik&rft.aufirst=Zolt%C3%A1n&rft.au=Lei%C3%9F%2C+Hans&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemiring" class="Z3988"> </li> <li id="cite_note-16"><a href="#cite_ref-16">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLehmann1977" class="citation cs2">Lehmann, Daniel J. (1977), <a rel="nofollow" class="external text" href="http://wrap.warwick.ac.uk/46308/7/WRAP_Lehmann_cs-rr-010.pdf">"Algebraic structures for transitive closure"</a> (PDF), Theoretical Computer Science, 4 (1): 59–76, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0304-3975%2877%2990056-1">10.1016/0304-3975(77)90056-1</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Theoretical+Computer+Science&rft.atitle=Algebraic+structures+for+transitive+closure&rft.volume=4&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E59-%3C%2Fspan%3E76&rft.date=1977&rft_id=info%3Adoi%2F10.1016%2F0304-3975%2877%2990056-1&rft.aulast=Lehmann&rft.aufirst=Daniel+J.&rft_id=http%3A%2F%2Fwrap.warwick.ac.uk%2F46308%2F7%2FWRAP_Lehmann_cs-rr-010.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemiring" class="Z3988"> </li> <li id="cite_note-FOOTNOTEBerstelReutenauer201127-17"><a href="#cite_ref-FOOTNOTEBerstelReutenauer201127_17-0">^</a> <a href="#CITEREFBerstelReutenauer2011">Berstel & Reutenauer (2011)</a>, p. 27 </li> <li id="cite_note-18"><a href="#cite_ref-18">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFÉsikKuich2004" class="citation book cs1">Ésik, Zoltán; Kuich, Werner (2004). "Equational axioms for a theory of automata". In Martín-Vide, Carlos (ed.). Formal languages and applications. Studies in Fuzziness and Soft Computing. Vol. 148. Berlin: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. pp. 183–196. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-20907-7" title="Special:BookSources/3-540-20907-7"><bdi>3-540-20907-7</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1088.68117">1088.68117</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Equational+axioms+for+a+theory+of+automata&rft.btitle=Formal+languages+and+applications&rft.place=Berlin&rft.series=Studies+in+Fuzziness+and+Soft+Computing&rft.pages=%3Cspan+class%3D%22nowrap%22%3E183-%3C%2Fspan%3E196&rft.pub=Springer-Verlag&rft.date=2004&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1088.68117%23id-name%3DZbl&rft.isbn=3-540-20907-7&rft.aulast=%C3%89sik&rft.aufirst=Zolt%C3%A1n&rft.au=Kuich%2C+Werner&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemiring" class="Z3988"> </li> <li id="cite_note-FOOTNOTEDrosteKuich200915Theorem_3.4-19"><a href="#cite_ref-FOOTNOTEDrosteKuich200915Theorem_3.4_19-0">^</a> <a href="#CITEREFDrosteKuich2009">Droste & Kuich (2009)</a>, p. 15, Theorem 3.4 </li> <li id="cite_note-20"><a href="#cite_ref-20">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFConway1971" class="citation book cs1"><a href="/wiki/John_Horton_Conway" title="John Horton Conway">Conway, J.H.</a> (1971). Regular algebra and finite machines. London: Chapman and Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-412-10620-5" title="Special:BookSources/0-412-10620-5"><bdi>0-412-10620-5</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0231.94041">0231.94041</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Regular+algebra+and+finite+machines&rft.place=London&rft.pub=Chapman+and+Hall&rft.date=1971&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0231.94041%23id-name%3DZbl&rft.isbn=0-412-10620-5&rft.aulast=Conway&rft.aufirst=J.H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemiring" class="Z3988"> </li> <li id="cite_note-Gut08-21">^ <a href="#cite_ref-Gut08_21-0">a</a> <a href="#cite_ref-Gut08_21-1">b</a> <a href="#cite_ref-Gut08_21-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGuterman2008" class="citation book cs1">Guterman, Alexander E. (2008). "Rank and determinant functions for matrices over semirings". In Young, Nicholas; Choi, Yemon (eds.). Surveys in Contemporary Mathematics. London Mathematical Society Lecture Note Series. Vol. 347. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. pp. 1–33. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-70564-6" title="Special:BookSources/978-0-521-70564-6"><bdi>978-0-521-70564-6</bdi></a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0076-0552">0076-0552</a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1181.16042">1181.16042</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Rank+and+determinant+functions+for+matrices+over+semirings&rft.btitle=Surveys+in+Contemporary+Mathematics&rft.series=London+Mathematical+Society+Lecture+Note+Series&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1-%3C%2Fspan%3E33&rft.pub=Cambridge+University+Press&rft.date=2008&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1181.16042%23id-name%3DZbl&rft.issn=0076-0552&rft.isbn=978-0-521-70564-6&rft.aulast=Guterman&rft.aufirst=Alexander+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemiring" class="Z3988"> </li> <li id="cite_note-FOOTNOTESakarovitch200928-22">^ <a href="#cite_ref-FOOTNOTESakarovitch200928_22-0">a</a> <a href="#cite_ref-FOOTNOTESakarovitch200928_22-1">b</a> <a href="#cite_ref-FOOTNOTESakarovitch200928_22-2">c</a> <a href="#CITEREFSakarovitch2009">Sakarovitch (2009)</a>, p. 28. </li> <li id="cite_note-FOOTNOTEBerstelReutenauer20114-23">^ <a href="#cite_ref-FOOTNOTEBerstelReutenauer20114_23-0">a</a> <a href="#cite_ref-FOOTNOTEBerstelReutenauer20114_23-1">b</a> <a href="#cite_ref-FOOTNOTEBerstelReutenauer20114_23-2">c</a> <a href="#CITEREFBerstelReutenauer2011">Berstel & Reutenauer (2011)</a>, p. 4 </li> <li id="cite_note-24"><a href="#cite_ref-24">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSpeyerSturmfels2009" class="citation journal cs1">Speyer, David; <a href="/wiki/Bernd_Sturmfels" title="Bernd Sturmfels">Sturmfels, Bernd</a> (2009) [2004]. "Tropical Mathematics". Math. Mag. 82 (3): 163–173. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0408099">math/0408099</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4169%2F193009809x468760">10.4169/193009809x468760</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119142649">119142649</a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1227.14051">1227.14051</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Math.+Mag.&rft.atitle=Tropical+Mathematics&rft.volume=82&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E163-%3C%2Fspan%3E173&rft.date=2009&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1227.14051%23id-name%3DZbl&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119142649%23id-name%3DS2CID&rft_id=info%3Aarxiv%2Fmath%2F0408099&rft_id=info%3Adoi%2F10.4169%2F193009809x468760&rft.aulast=Speyer&rft.aufirst=David&rft.au=Sturmfels%2C+Bernd&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemiring" class="Z3988"> </li> <li id="cite_note-25"><a href="#cite_ref-25">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSpeyerSturmfels2009" class="citation journal cs1">Speyer, David; Sturmfels, Bernd (2009). <a rel="nofollow" class="external text" href="https://www.tandfonline.com/doi/full/10.1080/0025570X.2009.11953615">"Tropical Mathematics"</a>. Mathematics Magazine. 82 (3): 163–173. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F0025570X.2009.11953615">10.1080/0025570X.2009.11953615</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0025-570X">0025-570X</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:15278805">15278805</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+Magazine&rft.atitle=Tropical+Mathematics&rft.volume=82&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E163-%3C%2Fspan%3E173&rft.date=2009&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A15278805%23id-name%3DS2CID&rft.issn=0025-570X&rft_id=info%3Adoi%2F10.1080%2F0025570X.2009.11953615&rft.aulast=Speyer&rft.aufirst=David&rft.au=Sturmfels%2C+Bernd&rft_id=https%3A%2F%2Fwww.tandfonline.com%2Fdoi%2Ffull%2F10.1080%2F0025570X.2009.11953615&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemiring" class="Z3988"> </li> <li id="cite_note-26"><a href="#cite_ref-26">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFJohn_C._Baez2001" class="citation newsgroup cs1"><a href="/wiki/John_C._Baez" title="John C. Baez">John C. Baez</a> (6 Nov 2001). <a rel="nofollow" class="external text" href="https://groups.google.com/d/msg/sci.physics.research/VJNPMCfreao/TMKt9tFYNwEJ">"quantum mechanics over a commutative rig"</a>. <a href="/wiki/Usenet_newsgroup" title="Usenet newsgroup">Newsgroup</a>: <a rel="nofollow" class="external text" href="news:sci.physics.research">sci.physics.research</a>. <a href="/wiki/Usenet_(identifier)" class="mw-redirect" title="Usenet (identifier)">Usenet:</a> <a rel="nofollow" class="external text" href="news:9s87n0$iv5@gap.cco.caltech.edu">9s87n0$iv5@gap.cco.caltech.edu</a>. Retrieved November 25, 2018.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=quantum+mechanics+over+a+commutative+rig&rft.pub=sci.physics.research&rft.date=2001-11-06&rft_id=news%3A9s87n0%24iv5%40gap.cco.caltech.edu%23id-name%3DUsenet%3A&rft.au=John+C.+Baez&rft_id=https%3A%2F%2Fgroups.google.com%2Fd%2Fmsg%2Fsci.physics.research%2FVJNPMCfreao%2FTMKt9tFYNwEJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemiring" class="Z3988"> </li> <li id="cite_note-27"><a href="#cite_ref-27">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBard2009" class="citation cs2">Bard, Gregory V. (2009), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=kjbp0mgu3IAC&pg=PA30">Algebraic Cryptanalysis</a>, Springer, Section 4.2.1, "Combinatorial Classes", ff., pp. 30–34, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780387887579" title="Special:BookSources/9780387887579"><bdi>9780387887579</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebraic+Cryptanalysis&rft.pages=Section+4.2.1%2C+%22Combinatorial+Classes%22%2C+ff.%2C+pp.+30-34&rft.pub=Springer&rft.date=2009&rft.isbn=9780387887579&rft.aulast=Bard&rft.aufirst=Gregory+V.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dkjbp0mgu3IAC%26pg%3DPA30&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemiring" class="Z3988"> </li> <li id="cite_note-28"><a href="#cite_ref-28">^</a> Schanuel S.H. (1991) Negative sets have Euler characteristic and dimension. In: Carboni A., Pedicchio M.C., Rosolini G. (eds) Category Theory. Lecture Notes in Mathematics, vol 1488. Springer, Berlin, Heidelberg </li> <li id="cite_note-FOOTNOTEPair1967271-30"><a href="#cite_ref-FOOTNOTEPair1967271_30-0">^</a> <a href="#CITEREFPair1967">Pair (1967)</a>, p. 271. </li> <li id="cite_note-FOOTNOTEDerniamePair1971-31"><a href="#cite_ref-FOOTNOTEDerniamePair1971_31-0">^</a> <a href="#CITEREFDerniamePair1971">Derniame & Pair (1971)</a> </li> <li id="cite_note-FOOTNOTEGolan19991Ch_1-32"><a href="#cite_ref-FOOTNOTEGolan19991Ch_1_32-0">^</a> <a href="#CITEREFGolan1999">Golan (1999)</a>, p. 1, Ch 1 </li> <li id="cite_note-FOOTNOTEGondranMinoux200822Ch_1,_§4.2-33"><a href="#cite_ref-FOOTNOTEGondranMinoux200822Ch_1,_§4.2_33-0">^</a> <a href="#CITEREFGondranMinoux2008">Gondran & Minoux (2008)</a>, p. 22, Ch 1, §4.2. </li> <li id="cite_note-FOOTNOTEGondranMinoux200820Ch_1,_§4.1-34"><a href="#cite_ref-FOOTNOTEGondranMinoux200820Ch_1,_§4.1_34-0">^</a> <a href="#CITEREFGondranMinoux2008">Gondran & Minoux (2008)</a>, p. 20, Ch 1, §4.1. </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2>[<a href="/w/index.php?title=Semiring&action=edit&section=27" title="Edit section: Bibliography">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDerniamePair1971" class="citation cs2">Derniame, Jean Claude; Pair, Claude (1971), Problèmes de cheminement dans les graphes (Path Problems in Graphs), Paris: Dunod</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Probl%C3%A8mes+de+cheminement+dans+les+graphes+%28Path+Problems+in+Graphs%29&rft.place=Paris&rft.pub=Dunod&rft.date=1971&rft.aulast=Derniame&rft.aufirst=Jean+Claude&rft.au=Pair%2C+Claude&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemiring" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBaccelliCohenOlsderQuadrat1992" class="citation cs2"><a href="/wiki/Fran%C3%A7ois_Baccelli" title="François Baccelli">Baccelli, François</a>; Cohen, Guy; Olsder, Geert Jan; Quadrat, Jean-Pierre (1992), <a rel="nofollow" class="external text" href="http://cermics.enpc.fr/~cohen-g//SED/book-online.html">Synchronization and Linearity (online version)</a>, Wiley, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-93609-X" title="Special:BookSources/0-471-93609-X"><bdi>0-471-93609-X</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Synchronization+and+Linearity+%28online+version%29&rft.pub=Wiley&rft.date=1992&rft.isbn=0-471-93609-X&rft.aulast=Baccelli&rft.aufirst=Fran%C3%A7ois&rft.au=Cohen%2C+Guy&rft.au=Olsder%2C+Geert+Jan&rft.au=Quadrat%2C+Jean-Pierre&rft_id=http%3A%2F%2Fcermics.enpc.fr%2F~cohen-g%2F%2FSED%2Fbook-online.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemiring" class="Z3988"></li> <li>Golan, Jonathan S. (1999) Semirings and their applications. Updated and expanded version of The theory of semirings, with applications to mathematics and theoretical computer science (Longman Sci. Tech., Harlow, 1992, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1163371">1163371</a>). Kluwer Academic Publishers, Dordrecht. xii+381 pp. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7923-5786-8" title="Special:BookSources/0-7923-5786-8">0-7923-5786-8</a> <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1746739">1746739</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBerstelPerrin1985" class="citation book cs1">Berstel, Jean; Perrin, Dominique (1985). Theory of codes. Pure and applied mathematics. Vol. 117. Academic Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-093420-1" title="Special:BookSources/978-0-12-093420-1"><bdi>978-0-12-093420-1</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0587.68066">0587.68066</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+of+codes&rft.series=Pure+and+applied+mathematics&rft.pub=Academic+Press&rft.date=1985&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0587.68066%23id-name%3DZbl&rft.isbn=978-0-12-093420-1&rft.aulast=Berstel&rft.aufirst=Jean&rft.au=Perrin%2C+Dominique&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemiring" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBerstelReutenauer2011" class="citation book cs1">Berstel, Jean; Reutenauer, Christophe (2011). Noncommutative rational series with applications. Encyclopedia of Mathematics and Its Applications. Vol. 137. Cambridge: <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-19022-0" title="Special:BookSources/978-0-521-19022-0"><bdi>978-0-521-19022-0</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1250.68007">1250.68007</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Noncommutative+rational+series+with+applications&rft.place=Cambridge&rft.series=Encyclopedia+of+Mathematics+and+Its+Applications&rft.pub=Cambridge+University+Press&rft.date=2011&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1250.68007%23id-name%3DZbl&rft.isbn=978-0-521-19022-0&rft.aulast=Berstel&rft.aufirst=Jean&rft.au=Reutenauer%2C+Christophe&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemiring" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDrosteKuich2009" class="citation cs2">Droste, Manfred; Kuich, Werner (2009), "Chapter 1: Semirings and Formal Power Series", Handbook of Weighted Automata, pp. 3–28, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-01492-5_1">10.1007/978-3-642-01492-5_1</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+1%3A+Semirings+and+Formal+Power+Series&rft.btitle=Handbook+of+Weighted+Automata&rft.pages=%3Cspan+class%3D%22nowrap%22%3E3-%3C%2Fspan%3E28&rft.date=2009&rft_id=info%3Adoi%2F10.1007%2F978-3-642-01492-5_1&rft.aulast=Droste&rft.aufirst=Manfred&rft.au=Kuich%2C+Werner&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemiring" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDurrett2019" class="citation book cs1"><a href="/wiki/Richard_Durrett" class="mw-redirect" title="Richard Durrett">Durrett, Richard</a> (2019). <a rel="nofollow" class="external text" href="https://services.math.duke.edu/~rtd/PTE/PTE5_011119.pdf">Probability: Theory and Examples</a> (PDF). Cambridge Series in Statistical and Probabilistic Mathematics. Vol. 49 (5th ed.). Cambridge New York, NY: <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-108-47368-2" title="Special:BookSources/978-1-108-47368-2"><bdi>978-1-108-47368-2</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1100115281">1100115281</a>. Retrieved November 5, 2020.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Probability%3A+Theory+and+Examples&rft.place=Cambridge+New+York%2C+NY&rft.series=Cambridge+Series+in+Statistical+and+Probabilistic+Mathematics&rft.edition=5th&rft.pub=Cambridge+University+Press&rft.date=2019&rft_id=info%3Aoclcnum%2F1100115281&rft.isbn=978-1-108-47368-2&rft.aulast=Durrett&rft.aufirst=Richard&rft_id=https%3A%2F%2Fservices.math.duke.edu%2F~rtd%2FPTE%2FPTE5_011119.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemiring" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFFolland1999" class="citation cs2">Folland, Gerald B. (1999), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=N8jVDwAAQBAJ&pg=PA23">Real Analysis: Modern Techniques and Their Applications</a> (2nd ed.), John Wiley & Sons, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780471317166" title="Special:BookSources/9780471317166"><bdi>9780471317166</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Real+Analysis%3A+Modern+Techniques+and+Their+Applications&rft.edition=2nd&rft.pub=John+Wiley+%26+Sons&rft.date=1999&rft.isbn=9780471317166&rft.aulast=Folland&rft.aufirst=Gerald+B.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DN8jVDwAAQBAJ%26pg%3DPA23&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemiring" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGolan1999" class="citation cs2">Golan, Jonathan S. (1999), Semirings and their Applications, Dordrecht: Kluwer Academic Publishers, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-94-015-9333-5">10.1007/978-94-015-9333-5</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7923-5786-8" title="Special:BookSources/0-7923-5786-8"><bdi>0-7923-5786-8</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1746739">1746739</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Semirings+and+their+Applications&rft.place=Dordrecht&rft.pub=Kluwer+Academic+Publishers&rft.date=1999&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1746739%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-94-015-9333-5&rft.isbn=0-7923-5786-8&rft.aulast=Golan&rft.aufirst=Jonathan+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemiring" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLothaire2005" class="citation book cs1"><a href="/wiki/M._Lothaire" title="M. Lothaire">Lothaire, M.</a> (2005). <a rel="nofollow" class="external text" href="https://archive.org/details/appliedcombinato0000loth">Applied combinatorics on words</a>. Encyclopedia of Mathematics and Its Applications. Vol. 105. A collective work by Jean Berstel, Dominique Perrin, Maxime Crochemore, Eric Laporte, Mehryar Mohri, Nadia Pisanti, Marie-France Sagot, <a href="/wiki/Gesine_Reinert" title="Gesine Reinert">Gesine Reinert</a>, <a href="/wiki/Sophie_Schbath" title="Sophie Schbath">Sophie Schbath</a>, Michael Waterman, Philippe Jacquet, <a href="/wiki/Wojciech_Szpankowski" title="Wojciech Szpankowski">Wojciech Szpankowski</a>, Dominique Poulalhon, Gilles Schaeffer, Roman Kolpakov, Gregory Koucherov, Jean-Paul Allouche and <a href="/wiki/Val%C3%A9rie_Berth%C3%A9" title="Valérie Berthé">Valérie Berthé</a>. Cambridge: <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-84802-4" title="Special:BookSources/0-521-84802-4"><bdi>0-521-84802-4</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1133.68067">1133.68067</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Applied+combinatorics+on+words&rft.place=Cambridge&rft.series=Encyclopedia+of+Mathematics+and+Its+Applications&rft.pub=Cambridge+University+Press&rft.date=2005&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1133.68067%23id-name%3DZbl&rft.isbn=0-521-84802-4&rft.aulast=Lothaire&rft.aufirst=M.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fappliedcombinato0000loth&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemiring" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGłazek2002" class="citation book cs1">Głazek, Kazimierz (2002). A guide to the literature on semirings and their applications in mathematics and information sciences. With complete bibliography. Dordrecht: Kluwer Academic. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-4020-0717-5" title="Special:BookSources/1-4020-0717-5"><bdi>1-4020-0717-5</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1072.16040">1072.16040</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+guide+to+the+literature+on+semirings+and+their+applications+in+mathematics+and+information+sciences.+With+complete+bibliography&rft.place=Dordrecht&rft.pub=Kluwer+Academic&rft.date=2002&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1072.16040%23id-name%3DZbl&rft.isbn=1-4020-0717-5&rft.aulast=G%C5%82azek&rft.aufirst=Kazimierz&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemiring" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGondranMinoux2008" class="citation book cs1">Gondran, Michel; Minoux, Michel (2008). Graphs, Dioids and Semirings: New Models and Algorithms. Operations Research/Computer Science Interfaces Series. Vol. 41. Dordrecht: Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-75450-5" title="Special:BookSources/978-0-387-75450-5"><bdi>978-0-387-75450-5</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1201.16038">1201.16038</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Graphs%2C+Dioids+and+Semirings%3A+New+Models+and+Algorithms&rft.place=Dordrecht&rft.series=Operations+Research%2FComputer+Science+Interfaces+Series&rft.pub=Springer+Science+%26+Business+Media&rft.date=2008&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1201.16038%23id-name%3DZbl&rft.isbn=978-0-387-75450-5&rft.aulast=Gondran&rft.aufirst=Michel&rft.au=Minoux%2C+Michel&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemiring" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPair1967" class="citation cs2">Pair, Claude (1967), "Sur des algorithmes pour des problèmes de cheminement dans les graphes finis (On algorithms for path problems in finite graphs)", in Rosentiehl (ed.), Théorie des graphes (journées internationales d'études) – Theory of Graphs (international symposium), Rome (Italy), July 1966: Dunod (Paris) et Gordon and Breach (New York)</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Sur+des+algorithmes+pour+des+probl%C3%A8mes+de+cheminement+dans+les+graphes+finis+%28On+algorithms+for+path+problems+in+finite+graphs%29&rft.btitle=Th%C3%A9orie+des+graphes+%28journ%C3%A9es+internationales+d%27%C3%A9tudes%29+%E2%80%93+Theory+of+Graphs+%28international+symposium%29&rft.place=Rome+%28Italy%29%2C+July+1966&rft.pub=Dunod+%28Paris%29+et+Gordon+and+Breach+%28New+York%29&rft.date=1967&rft.aulast=Pair&rft.aufirst=Claude&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemiring" class="Z3988"><code class="cs1-code">{{<a href="/wiki/Template:Citation" title="Template:Citation">citation</a>}}</code>: CS1 maint: location (<a href="/wiki/Category:CS1_maint:_location" title="Category:CS1 maint: location">link</a>)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSakarovitch2009" class="citation book cs1">Sakarovitch, Jacques (2009). Elements of automata theory. Translated from the French by Reuben Thomas. Cambridge: <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-84425-3" title="Special:BookSources/978-0-521-84425-3"><bdi>978-0-521-84425-3</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1188.68177">1188.68177</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elements+of+automata+theory&rft.place=Cambridge&rft.pub=Cambridge+University+Press&rft.date=2009&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1188.68177%23id-name%3DZbl&rft.isbn=978-0-521-84425-3&rft.aulast=Sakarovitch&rft.aufirst=Jacques&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemiring" class="Z3988"></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2>[<a href="/w/index.php?title=Semiring&action=edit&section=28" title="Edit section: Further reading">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239549316" /><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGolan2003" class="citation book cs1">Golan, Jonathan S. (2003). Semirings and Affine Equations over Them. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4020-1358-4" title="Special:BookSources/978-1-4020-1358-4"><bdi>978-1-4020-1358-4</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1042.16038">1042.16038</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Semirings+and+Affine+Equations+over+Them&rft.pub=Springer+Science+%26+Business+Media&rft.date=2003&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1042.16038%23id-name%3DZbl&rft.isbn=978-1-4020-1358-4&rft.aulast=Golan&rft.aufirst=Jonathan+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemiring" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGrillet1970" class="citation journal cs1">Grillet, Mireille P. (1970). <a rel="nofollow" class="external text" href="https://eudml.org/doc/115127">"Green's relations in a semiring"</a>. Port. Math. 29: 181–195. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0227.16029">0227.16029</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Port.+Math.&rft.atitle=Green%27s+relations+in+a+semiring&rft.volume=29&rft.pages=%3Cspan+class%3D%22nowrap%22%3E181-%3C%2Fspan%3E195&rft.date=1970&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0227.16029%23id-name%3DZbl&rft.aulast=Grillet&rft.aufirst=Mireille+P.&rft_id=https%3A%2F%2Feudml.org%2Fdoc%2F115127&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemiring" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGunawardena1998" class="citation book cs1">Gunawardena, Jeremy (1998). "An introduction to idempotency". In Gunawardena, Jeremy (ed.). <a rel="nofollow" class="external text" href="http://www.hpl.hp.com/techreports/96/HPL-BRIMS-96-24.pdf">Idempotency. Based on a workshop, Bristol, UK, October 3–7, 1994</a> (PDF). Cambridge: <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. pp. 1–49. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0898.16032">0898.16032</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=An+introduction+to+idempotency&rft.btitle=Idempotency.+Based+on+a+workshop%2C+Bristol%2C+UK%2C+October+3%E2%80%937%2C+1994&rft.place=Cambridge&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1-%3C%2Fspan%3E49&rft.pub=Cambridge+University+Press&rft.date=1998&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0898.16032%23id-name%3DZbl&rft.aulast=Gunawardena&rft.aufirst=Jeremy&rft_id=http%3A%2F%2Fwww.hpl.hp.com%2Ftechreports%2F96%2FHPL-BRIMS-96-24.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemiring" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFJipsen2004" class="citation journal cs1">Jipsen, P. (2004). "From semirings to residuated Kleene lattices". Studia Logica. 76 (2): 291–303. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1023%2FB%3ASTUD.0000032089.54776.63">10.1023/B:STUD.0000032089.54776.63</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:9946523">9946523</a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1045.03049">1045.03049</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Studia+Logica&rft.atitle=From+semirings+to+residuated+Kleene+lattices&rft.volume=76&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E291-%3C%2Fspan%3E303&rft.date=2004&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1045.03049%23id-name%3DZbl&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A9946523%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1023%2FB%3ASTUD.0000032089.54776.63&rft.aulast=Jipsen&rft.aufirst=P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemiring" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDolan2013" class="citation cs2">Dolan, Steven (2013), <a rel="nofollow" class="external text" href="http://www.cl.cam.ac.uk/~sd601/papers/semirings.pdf">"Fun with Semirings"</a> (PDF), Proceedings of the 18th ACM SIGPLAN international conference on Functional programming, 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