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id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Boolean algebra (structure)</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" 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Available in 18 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-18" aria-hidden="true" > 18 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%91%D1%83%D0%BB%D1%8C_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D2%BB%D1%8B" title="Буль алгебраһы – Bashkir" lang="ba" hreflang="ba" data-title="Буль алгебраһы" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target">Башҡортса</a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%91%D1%83%D0%BB%D1%8C_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B8" title="Буль алгебри – Chuvash" lang="cv" hreflang="cv" data-title="Буль алгебри" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target">Чӑвашла</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Alg%C3%A8bre_de_Boole_(structure)" title="Algèbre de Boole (structure) – French" lang="fr" hreflang="fr" data-title="Algèbre de Boole (structure)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B6%88_%EB%8C%80%EC%88%98" title="불 대수 – Korean" lang="ko" hreflang="ko" data-title="불 대수" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Algebro_di_Boole" title="Algebro di Boole – Ido" lang="io" hreflang="io" data-title="Algebro di Boole" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target">Ido</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Aljabar_Boolean_(struktur)" title="Aljabar Boolean (struktur) – Indonesian" lang="id" hreflang="id" data-title="Aljabar Boolean (struktur)" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94_%D7%91%D7%95%D7%9C%D7%99%D7%90%D7%A0%D7%99%D7%AA_(%D7%9E%D7%91%D7%A0%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99)" title="אלגברה בוליאנית (מבנה אלגברי) – Hebrew" lang="he" hreflang="he" data-title="אלגברה בוליאנית (מבנה אלגברי)" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%91%D1%83%D0%BB%D1%8C_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D1%81%D1%8B" title="Буль алгебрасы – Kazakh" lang="kk" hreflang="kk" data-title="Буль алгебрасы" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Alj%C3%A8b_Boole_(estrikti)" title="Aljèb Boole (estrikti) – Haitian Creole" lang="ht" hreflang="ht" data-title="Aljèb Boole (estrikti)" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target">Kreyòl ayisyen</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Boole-algebra_(strukt%C3%BAra)" title="Boole-algebra (struktúra) – Hungarian" lang="hu" hreflang="hu" data-title="Boole-algebra (struktúra)" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%96%E3%83%BC%E3%83%AB%E4%BB%A3%E6%95%B0" title="ブール代数 – Japanese" lang="ja" hreflang="ja" data-title="ブール代数" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Mantiqiy_amallar" title="Mantiqiy amallar – Uzbek" lang="uz" hreflang="uz" data-title="Mantiqiy amallar" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target">Oʻzbekcha / ўзбекча</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Algebra_Boole%E2%80%99a" title="Algebra Boole’a – Polish" lang="pl" hreflang="pl" data-title="Algebra Boole’a" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/%C3%81lgebra_booliana_(estrutura)" title="Álgebra booliana (estrutura) – Portuguese" lang="pt" hreflang="pt" data-title="Álgebra booliana (estrutura)" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%91%D1%83%D0%BB%D0%B5%D0%B2%D0%B0_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Булева алгебра – Russian" lang="ru" hreflang="ru" data-title="Булева алгебра" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%91%D1%83%D0%BB%D0%B5%D0%B2%D0%B0_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0_(%D1%81%D1%82%D1%80%D1%83%D0%BA%D1%82%D1%83%D1%80%D0%B0)" title="Булева алгебра (структура) – Ukrainian" lang="uk" hreflang="uk" data-title="Булева алгебра (структура)" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%B8%83%E5%B0%94%E4%BB%A3%E6%95%B0" title="布尔代数 – Wu" lang="wuu" hreflang="wuu" data-title="布尔代数" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target">吴语</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%B8%83%E5%B0%94%E4%BB%A3%E6%95%B0" title="布尔代数 – Chinese" lang="zh" hreflang="zh" data-title="布尔代数" data-language-autonym="中文" data-language-local-name="Chinese" 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.hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For an introduction to the subject, see <a href="/wiki/Boolean_algebra" title="Boolean algebra">Boolean algebra</a>. For an alternative presentation, see <a href="/wiki/Boolean_algebras_canonically_defined" title="Boolean algebras canonically defined">Boolean algebras canonically defined</a>.</div> In <a href="/wiki/Abstract_algebra" title="Abstract algebra">abstract algebra</a>, a Boolean algebra or Boolean lattice is a <a href="/wiki/Complemented_lattice" title="Complemented lattice">complemented</a> <a href="/wiki/Distributive_lattice" title="Distributive lattice">distributive lattice</a>. This type of <a href="/wiki/Algebraic_structure" title="Algebraic structure">algebraic structure</a> captures essential properties of both <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> operations and <a href="/wiki/Logic" title="Logic">logic</a> operations. A Boolean algebra can be seen as a generalization of a <a href="/wiki/Power_set" title="Power set">power set</a> algebra or a <a href="/wiki/Field_of_sets" title="Field of sets">field of sets</a>, or its elements can be viewed as generalized <a href="/wiki/Truth_value" title="Truth value">truth values</a>. It is also a special case of a <a href="/wiki/De_Morgan_algebra" title="De Morgan algebra">De Morgan algebra</a> and a <a href="/wiki/Kleene_algebra_(with_involution)" class="mw-redirect" title="Kleene algebra (with involution)">Kleene algebra (with involution)</a>. Every Boolean algebra <a href="#Boolean_rings">gives rise</a> to a <a href="/wiki/Boolean_ring" title="Boolean ring">Boolean ring</a>, and vice versa, with <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a> multiplication corresponding to <a href="/wiki/Logical_conjunction" title="Logical conjunction">conjunction</a> or <a href="/wiki/Meet_(mathematics)" class="mw-redirect" title="Meet (mathematics)">meet</a> ∧, and ring addition to <a href="/wiki/Exclusive_or" title="Exclusive or">exclusive disjunction</a> or <a href="/wiki/Symmetric_difference" title="Symmetric difference">symmetric difference</a> (not <a href="/wiki/Logical_disjunction" title="Logical disjunction">disjunction</a> ∨). However, the theory of Boolean rings has an inherent asymmetry between the two operators, while the <a href="/wiki/Axiom" title="Axiom">axioms</a> and theorems of Boolean algebra express the symmetry of the theory described by the <a href="/wiki/Duality_principle_(Boolean_algebra)" class="mw-redirect" title="Duality principle (Boolean algebra)">duality principle</a>.<a href="#cite_note-FOOTNOTEGivantHalmos200920-1">[1]</a> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Hasse_diagram_of_powerset_of_3.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Hasse_diagram_of_powerset_of_3.svg/250px-Hasse_diagram_of_powerset_of_3.svg.png" decoding="async" width="250" height="189" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Hasse_diagram_of_powerset_of_3.svg/375px-Hasse_diagram_of_powerset_of_3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Hasse_diagram_of_powerset_of_3.svg/500px-Hasse_diagram_of_powerset_of_3.svg.png 2x" data-file-width="429" data-file-height="325" /></a><figcaption>Boolean lattice of subsets</figcaption></figure> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=Boolean_algebra_(structure)&action=edit&section=1" title="Edit section: History">edit</a>]</div> The term "Boolean algebra" honors <a href="/wiki/George_Boole" title="George Boole">George Boole</a> (1815–1864), a self-educated English mathematician. He introduced the <a href="/wiki/Algebraic_system" class="mw-redirect" title="Algebraic system">algebraic system</a> initially in a small pamphlet, The Mathematical Analysis of Logic, published in 1847 in response to an ongoing public controversy between <a href="/wiki/Augustus_De_Morgan" title="Augustus De Morgan">Augustus De Morgan</a> and <a href="/wiki/Sir_William_Hamilton,_9th_Baronet" title="Sir William Hamilton, 9th Baronet">William Hamilton</a>, and later as a more substantial book, <a href="/wiki/The_Laws_of_Thought" title="The Laws of Thought">The Laws of Thought</a>, published in 1854. Boole's formulation differs from that described above in some important respects. For example, conjunction and disjunction in Boole were not a dual pair of operations. Boolean algebra emerged in the 1860s, in papers written by <a href="/wiki/William_Jevons" class="mw-redirect" title="William Jevons">William Jevons</a> and <a href="/wiki/Charles_Sanders_Peirce" title="Charles Sanders Peirce">Charles Sanders Peirce</a>. The first systematic presentation of Boolean algebra and <a href="/wiki/Distributive_lattice" title="Distributive lattice">distributive lattices</a> is owed to the 1890 Vorlesungen of <a href="/wiki/Ernst_Schr%C3%B6der_(mathematician)" title="Ernst Schröder (mathematician)">Ernst Schröder</a>. The first extensive treatment of Boolean algebra in English is <a href="/wiki/A._N._Whitehead" class="mw-redirect" title="A. N. Whitehead">A. N. Whitehead</a>'s 1898 Universal Algebra. Boolean algebra as an axiomatic algebraic structure in the modern axiomatic sense begins with a 1904 paper by <a href="/wiki/Edward_V._Huntington" class="mw-redirect" title="Edward V. Huntington">Edward V. Huntington</a>. Boolean algebra came of age as serious mathematics with the work of <a href="/wiki/Marshall_Stone" class="mw-redirect" title="Marshall Stone">Marshall Stone</a> in the 1930s, and with <a href="/wiki/Garrett_Birkhoff" title="Garrett Birkhoff">Garrett Birkhoff</a>'s 1940 Lattice Theory. In the 1960s, <a href="/wiki/Paul_Cohen_(mathematician)" class="mw-redirect" title="Paul Cohen (mathematician)">Paul Cohen</a>, <a href="/wiki/Dana_Scott" title="Dana Scott">Dana Scott</a>, and others found deep new results in <a href="/wiki/Mathematical_logic" title="Mathematical logic">mathematical logic</a> and <a href="/wiki/Axiomatic_set_theory" class="mw-redirect" title="Axiomatic set theory">axiomatic set theory</a> using offshoots of Boolean algebra, namely <a href="/wiki/Forcing_(mathematics)" title="Forcing (mathematics)">forcing</a> and <a href="/wiki/Boolean-valued_model" title="Boolean-valued model">Boolean-valued models</a>. <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2>[<a href="/w/index.php?title=Boolean_algebra_(structure)&action=edit&section=2" title="Edit section: Definition">edit</a>]</div> A Boolean algebra is a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> A, equipped with two <a href="/wiki/Binary_operation" title="Binary operation">binary operations</a> ∧ (called "meet" or "and"), ∨ (called "join" or "or"), a <a href="/wiki/Unary_operation" title="Unary operation">unary operation</a> ¬ (called "complement" or "not") and two elements 0 and 1 in A (called "bottom" and "top", or "least" and "greatest" element, also denoted by the symbols ⊥ and ⊤, respectively), such that for all elements a, b and c of A, the following <a href="/wiki/Axiom" title="Axiom">axioms</a> hold:<a href="#cite_note-FOOTNOTEDaveyPriestley1990109,_131,_144-2">[2]</a> <dl><dd><dl><dd><table cellpadding="5"> <tbody><tr> <td>a ∨ (b ∨ c) = (a ∨ b) ∨ c </td> <td>a ∧ (b ∧ c) = (a ∧ b) ∧ c </td> <td><a href="/wiki/Associativity" class="mw-redirect" title="Associativity">associativity</a> </td></tr> <tr> <td>a ∨ b = b ∨ a </td> <td>a ∧ b = b ∧ a </td> <td><a href="/wiki/Commutativity" class="mw-redirect" title="Commutativity">commutativity</a> </td></tr> <tr> <td>a ∨ (a ∧ b) = a </td> <td>a ∧ (a ∨ b) = a </td> <td><a href="/wiki/Absorption_law" title="Absorption law">absorption</a> </td></tr> <tr> <td>a ∨ 0 = a </td> <td>a ∧ 1 = a </td> <td><a href="/wiki/Identity_element" title="Identity element">identity</a> </td></tr> <tr> <td>a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c)   </td> <td>a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c)   </td> <td><a href="/wiki/Distributivity" class="mw-redirect" title="Distributivity">distributivity</a> </td></tr> <tr> <td>a ∨ ¬a = 1 </td> <td>a ∧ ¬a = 0 </td> <td><a href="/wiki/Complemented_lattice" title="Complemented lattice">complements</a> </td></tr></tbody></table></dd></dl></dd></dl> Note, however, that the absorption law and even the associativity law can be excluded from the set of axioms as they can be derived from the other axioms (see <a href="#Axiomatics">Proven properties</a>). A Boolean algebra with only one element is called a trivial Boolean algebra or a degenerate Boolean algebra. (In older works, some authors required 0 and 1 to be distinct elements in order to exclude this case.)[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] It follows from the last three pairs of axioms above (identity, distributivity and complements), or from the absorption axiom, that <dl><dd>a = b ∧ a     if and only if     a ∨ b = b.</dd></dl> The relation ≤ defined by a ≤ b if these equivalent conditions hold, is a <a href="/wiki/Partial_order" class="mw-redirect" title="Partial order">partial order</a> with least element 0 and greatest element 1. The meet a ∧ b and the join a ∨ b of two elements coincide with their <a href="/wiki/Infimum" class="mw-redirect" title="Infimum">infimum</a> and <a href="/wiki/Supremum" class="mw-redirect" title="Supremum">supremum</a>, respectively, with respect to ≤. The first four pairs of axioms constitute a definition of a <a href="/wiki/Bounded_lattice" class="mw-redirect" title="Bounded lattice">bounded lattice</a>. It follows from the first five pairs of axioms that any complement is unique. The set of axioms is <a href="/wiki/Duality_(order_theory)" title="Duality (order theory)">self-dual</a> in the sense that if one exchanges ∨ with ∧ and 0 with 1 in an axiom, the result is again an axiom. Therefore, by applying this operation to a Boolean algebra (or Boolean lattice), one obtains another Boolean algebra with the same elements; it is called its dual.<a href="#cite_note-FOOTNOTEGoodstein201221ff-3">[3]</a> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2>[<a href="/w/index.php?title=Boolean_algebra_(structure)&action=edit&section=3" title="Edit section: Examples">edit</a>]</div> <ul><li>The simplest non-trivial Boolean algebra, the <a href="/wiki/Two-element_Boolean_algebra" title="Two-element Boolean algebra">two-element Boolean algebra</a>, has only two elements, 0 and 1, and is defined by the rules:</li></ul> <table> <tbody><tr> <td width="70"> </td> <td> <table class="wikitable"> <tbody><tr> <th>∧</th> <th>0</th> <th>1 </th></tr> <tr> <th>0 </th> <td>0</td> <td>0 </td></tr> <tr> <th>1 </th> <td>0</td> <td>1 </td></tr></tbody></table> </td> <td> </td> <td width="30"> </td> <td> <table class="wikitable"> <tbody><tr> <th>∨</th> <th>0</th> <th>1 </th></tr> <tr> <th>0 </th> <td>0</td> <td>1 </td></tr> <tr> <th>1 </th> <td>1</td> <td>1 </td></tr></tbody></table> </td> <td> </td> <td width="40"> </td> <td> <table class="wikitable"> <tbody><tr> <th>a</th> <th>0</th> <th>1 </th></tr> <tr> <th>¬a </th> <td>1</td> <td>0 </td></tr></tbody></table> </td></tr></tbody></table> <dl><dd><ul><li>It has applications in <a href="/wiki/Logic" title="Logic">logic</a>, interpreting 0 as false, 1 as true, ∧ as and, ∨ as or, and ¬ as not. Expressions involving variables and the Boolean operations represent statement forms, and two such expressions can be shown to be equal using the above axioms if and only if the corresponding statement forms are <a href="/wiki/Logical_equivalence" title="Logical equivalence">logically equivalent</a>.</li></ul></dd></dl> <dl><dd><ul><li>The two-element Boolean algebra is also used for circuit design in <a href="/wiki/Electrical_engineering" title="Electrical engineering">electrical engineering</a>;<a href="#cite_note-4">[note 1]</a> here 0 and 1 represent the two different states of one <a href="/wiki/Bit" title="Bit">bit</a> in a <a href="/wiki/Digital_circuit" class="mw-redirect" title="Digital circuit">digital circuit</a>, typically high and low <a href="/wiki/Voltage" title="Voltage">voltage</a>. Circuits are described by expressions containing variables, and two such expressions are equal for all values of the variables if and only if the corresponding circuits have the same input–output behavior. Furthermore, every possible input–output behavior can be modeled by a suitable Boolean expression.</li></ul></dd></dl> <dl><dd><ul><li>The two-element Boolean algebra is also important in the general theory of Boolean algebras, because an equation involving several variables is generally true in all Boolean algebras if and only if it is true in the two-element Boolean algebra (which can be checked by a trivial <a href="/wiki/Brute_force_search" class="mw-redirect" title="Brute force search">brute force</a> algorithm for small numbers of variables). This can for example be used to show that the following laws (<a href="/wiki/Consensus_theorem" title="Consensus theorem">Consensus theorems</a>) are generally valid in all Boolean algebras: <ul><li>(a ∨ b) ∧ (¬a ∨ c) ∧ (b ∨ c) ≡ (a ∨ b) ∧ (¬a ∨ c)</li> <li>(a ∧ b) ∨ (¬a ∧ c) ∨ (b ∧ c) ≡ (a ∧ b) ∨ (¬a ∧ c)</li></ul></li></ul></dd></dl> <ul><li>The <a href="/wiki/Power_set" title="Power set">power set</a> (set of all subsets) of any given nonempty set S forms a Boolean algebra, an <a href="/wiki/Algebra_of_sets" title="Algebra of sets">algebra of sets</a>, with the two operations ∨ := ∪ (union) and ∧ := ∩ (intersection). The smallest element 0 is the <a href="/wiki/Empty_set" title="Empty set">empty set</a> and the largest element 1 is the set S itself.</li></ul> <dl><dd><ul><li>After the two-element Boolean algebra, the simplest Boolean algebra is that defined by the <a href="/wiki/Power_set" title="Power set">power set</a> of two atoms:</li></ul></dd></dl> <table> <tbody><tr> <td width="70"> </td> <td> <table class="wikitable"> <tbody><tr> <th>∧</th> <th>0</th> <th>a</th> <th>b</th> <th>1 </th></tr> <tr> <th>0 </th> <td>0</td> <td>0</td> <td>0</td> <td>0 </td></tr> <tr> <th>a </th> <td>0</td> <td>a</td> <td>0</td> <td>a </td></tr> <tr> <th>b </th> <td>0</td> <td>0</td> <td>b</td> <td>b </td></tr> <tr> <th>1 </th> <td>0</td> <td>a</td> <td>b</td> <td>1 </td></tr></tbody></table> </td> <td> </td> <td width="30"> </td> <td> <table class="wikitable"> <tbody><tr> <th>∨</th> <th>0</th> <th>a</th> <th>b</th> <th>1 </th></tr> <tr> <th>0 </th> <td>0</td> <td>a</td> <td>b</td> <td>1 </td></tr> <tr> <th>a </th> <td>a</td> <td>a</td> <td>1</td> <td>1 </td></tr> <tr> <th>b </th> <td>b</td> <td>1</td> <td>b</td> <td>1 </td></tr> <tr> <th>1 </th> <td>1</td> <td>1</td> <td>1</td> <td>1 </td></tr></tbody></table> </td> <td> </td> <td width="40"> </td> <td> <table class="wikitable"> <tbody><tr> <th>x</th> <th>0</th> <th>a</th> <th>b</th> <th>1 </th></tr> <tr> <th>¬x </th> <td>1</td> <td>b</td> <td>a</td> <td>0 </td></tr></tbody></table> </td></tr></tbody></table> <ul><li>The set A of all subsets of S that are either finite or <a href="/wiki/Cofinite" class="mw-redirect" title="Cofinite">cofinite</a> is a Boolean algebra and an <a href="/wiki/Algebra_of_sets" title="Algebra of sets">algebra of sets</a> called the <a href="/wiki/Finite%E2%80%93cofinite_algebra" class="mw-redirect" title="Finite–cofinite algebra">finite–cofinite algebra</a>. If S is infinite then the set of all cofinite subsets of S, which is called the <a href="/wiki/Fr%C3%A9chet_filter" title="Fréchet filter">Fréchet filter</a>, is a free <a href="/wiki/Ultrafilter" title="Ultrafilter">ultrafilter</a> on A. However, the Fréchet filter is not an ultrafilter on the power set of S.</li> <li>Starting with the <a href="/wiki/Propositional_calculus" title="Propositional calculus">propositional calculus</a> with κ sentence symbols, form the <a href="/wiki/Lindenbaum%E2%80%93Tarski_algebra" title="Lindenbaum–Tarski algebra">Lindenbaum algebra</a> (that is, the set of sentences in the propositional calculus modulo <a href="/wiki/Logical_equivalence" title="Logical equivalence">logical equivalence</a>). This construction yields a Boolean algebra. It is in fact the <a href="/wiki/Free_Boolean_algebra" title="Free Boolean algebra">free Boolean algebra</a> on κ generators. A truth assignment in propositional calculus is then a Boolean algebra homomorphism from this algebra to the two-element Boolean algebra.</li> <li>Given any <a href="/wiki/Linearly_ordered" class="mw-redirect" title="Linearly ordered">linearly ordered</a> set L with a least element, the interval algebra is the smallest Boolean algebra of subsets of L containing all of the half-open intervals [a, b) such that a is in L and b is either in L or equal to ∞. Interval algebras are useful in the study of <a href="/wiki/Lindenbaum%E2%80%93Tarski_algebra" title="Lindenbaum–Tarski algebra">Lindenbaum–Tarski algebras</a>; every <a href="/wiki/Countable" class="mw-redirect" title="Countable">countable</a> Boolean algebra is isomorphic to an interval algebra.</li></ul> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Lattice_T_30.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Lattice_T_30.svg/150px-Lattice_T_30.svg.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Lattice_T_30.svg/225px-Lattice_T_30.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Lattice_T_30.svg/300px-Lattice_T_30.svg.png 2x" data-file-width="100" data-file-height="100" /></a><figcaption><a href="/wiki/Hasse_diagram" title="Hasse diagram">Hasse diagram</a> of the Boolean algebra of divisors of 30.</figcaption></figure> <ul><li>For any <a href="/wiki/Natural_number" title="Natural number">natural number</a> n, the set of all positive <a href="/wiki/Divisor" title="Divisor">divisors</a> of n, defining a ≤ b if a <a href="/wiki/Divides" class="mw-redirect" title="Divides">divides</a> b, forms a <a href="/wiki/Distributive_lattice" title="Distributive lattice">distributive lattice</a>. This lattice is a Boolean algebra if and only if n is <a href="/wiki/Square-free_integer" title="Square-free integer">square-free</a>. The bottom and the top elements of this Boolean algebra are the natural numbers 1 and n, respectively. The complement of a is given by n/a. The meet and the join of a and b are given by the <a href="/wiki/Greatest_common_divisor" title="Greatest common divisor">greatest common divisor</a> (gcd) and the <a href="/wiki/Least_common_multiple" title="Least common multiple">least common multiple</a> (lcm) of a and b, respectively. The ring addition a + b is given by lcm(a, b) / gcd(a, b). The picture shows an example for n = 30. As a counter-example, considering the non-square-free n = 60, the greatest common divisor of 30 and its complement 2 would be 2, while it should be the bottom element 1.</li> <li>Other examples of Boolean algebras arise from <a href="/wiki/Topology" title="Topology">topological spaces</a>: if X is a topological space, then the collection of all subsets of X that are <a href="/wiki/Clopen_set" title="Clopen set">both open and closed</a> forms a Boolean algebra with the operations ∨ := ∪ (union) and ∧ := ∩ (intersection).</li> <li>If R is an arbitrary ring then its set of <a href="/wiki/Central_idempotent" class="mw-redirect" title="Central idempotent">central idempotents</a>, which is the set</li></ul> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=\left\{e\in R:e^{2}=e{\text{ and }}ex=xe\;{\text{ for all }}\;x\in R\right\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mi>e</mi> <mo>∈</mo> <mi>R</mi> <mo>:</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mi>e</mi> <mi>x</mi> <mo>=</mo> <mi>x</mi> <mi>e</mi> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> for all </mtext> </mrow> <mspace width="thickmathspace" /> <mi>x</mi> <mo>∈</mo> <mi>R</mi> </mrow> <mo>}</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\left\{e\in R:e^{2}=e{\text{ and }}ex=xe\;{\text{ for all }}\;x\in R\right\},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d86d094a2367aff1dd95ed2506748fc2d2a38312" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:49.573ex; height:3.343ex;" alt="{\displaystyle A=\left\{e\in R:e^{2}=e{\text{ and }}ex=xe\;{\text{ for all }}\;x\in R\right\},}"> becomes a Boolean algebra when its operations are defined by e ∨ f := e + f − ef and e ∧ f := ef. <div class="mw-heading mw-heading2"><h2 id="Homomorphisms_and_isomorphisms">Homomorphisms and isomorphisms</h2>[<a href="/w/index.php?title=Boolean_algebra_(structure)&action=edit&section=4" title="Edit section: Homomorphisms and isomorphisms">edit</a>]</div> A <a href="/wiki/Homomorphism" title="Homomorphism">homomorphism</a> between two Boolean algebras A and B is a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> f : A → B such that for all a, b in A: <dl><dd>f(a ∨ b) = f(a) ∨ f(b),</dd> <dd>f(a ∧ b) = f(a) ∧ f(b),</dd> <dd>f(0) = 0,</dd> <dd>f(1) = 1.</dd></dl> It then follows that f(¬a) = ¬f(a) for all a in A. The <a href="/wiki/Class_(set_theory)" title="Class (set theory)">class</a> of all Boolean algebras, together with this notion of morphism, forms a <a href="/wiki/Full_subcategory" class="mw-redirect" title="Full subcategory">full subcategory</a> of the <a href="/wiki/Category_theory" title="Category theory">category</a> of lattices. An isomorphism between two Boolean algebras A and B is a homomorphism f : A → B with an inverse homomorphism, that is, a homomorphism g : B → A such that the <a href="/wiki/Function_composition" title="Function composition">composition</a> g ∘ f : A → A is the <a href="/wiki/Identity_function" title="Identity function">identity function</a> on A, and the composition f ∘ g : B → B is the identity function on B. A homomorphism of Boolean algebras is an isomorphism if and only if it is <a href="/wiki/Bijection" title="Bijection">bijective</a>. <div class="mw-heading mw-heading2"><h2 id="Boolean_rings">Boolean rings</h2>[<a href="/w/index.php?title=Boolean_algebra_(structure)&action=edit&section=5" title="Edit section: Boolean rings">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Boolean_ring" title="Boolean ring">Boolean ring</a></div> Every Boolean algebra (A, ∧, ∨) gives rise to a <a href="/wiki/Ring_(algebra)" class="mw-redirect" title="Ring (algebra)">ring</a> (A, +, ·) by defining a + b := (a ∧ ¬b) ∨ (b ∧ ¬a) = (a ∨ b) ∧ ¬(a ∧ b) (this operation is called <a href="/wiki/Symmetric_difference" title="Symmetric difference">symmetric difference</a> in the case of sets and <a href="/wiki/Truth_table#Exclusive_disjunction" title="Truth table">XOR</a> in the case of logic) and a · b := a ∧ b. The zero element of this ring coincides with the 0 of the Boolean algebra; the multiplicative identity element of the ring is the 1 of the Boolean algebra. This ring has the property that a · a = a for all a in A; rings with this property are called <a href="/wiki/Boolean_ring" title="Boolean ring">Boolean rings</a>. Conversely, if a Boolean ring A is given, we can turn it into a Boolean algebra by defining x ∨ y := x + y + (x · y) and x ∧ y := x · y.<a href="#cite_note-FOOTNOTEStone1936-5">[4]</a><a href="#cite_note-FOOTNOTEHsiang1985260-6">[5]</a> Since these two constructions are inverses of each other, we can say that every Boolean ring arises from a Boolean algebra, and vice versa. Furthermore, a map f : A → B is a homomorphism of Boolean algebras if and only if it is a homomorphism of Boolean rings. The <a href="/wiki/Category_theory" title="Category theory">categories</a> of Boolean rings and Boolean algebras are <a href="/wiki/Equivalence_of_categories" title="Equivalence of categories">equivalent</a>;<a href="#cite_note-FOOTNOTECohn2003[httpsbooksgooglecombooksidVESm0MJOiDQCpgPA81_81]-7">[6]</a> in fact the categories are <a href="/wiki/Isomorphism_of_categories" title="Isomorphism of categories">isomorphic</a>. Hsiang (1985) gave a <a href="/wiki/Abstract_rewriting_system" title="Abstract rewriting system">rule-based algorithm</a> to <a href="/wiki/Word_problem_(mathematics)" title="Word problem (mathematics)">check</a> whether two arbitrary expressions denote the same value in every Boolean ring. More generally, Boudet, <a href="/wiki/Jean-Pierre_Jouannaud" title="Jean-Pierre Jouannaud">Jouannaud</a>, and Schmidt-Schauß (1989) gave an algorithm to <a href="/wiki/Unification_(computer_science)#Particular_background_knowledge_sets_E" title="Unification (computer science)">solve equations</a> between arbitrary Boolean-ring expressions. Employing the similarity of Boolean rings and Boolean algebras, both algorithms have applications in <a href="/wiki/Automated_theorem_proving" title="Automated theorem proving">automated theorem proving</a>. <div class="mw-heading mw-heading2"><h2 id="Ideals_and_filters">Ideals and filters</h2>[<a href="/w/index.php?title=Boolean_algebra_(structure)&action=edit&section=6" title="Edit section: Ideals and filters">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Ideal_(order_theory)" title="Ideal (order theory)">Ideal (order theory)</a> and <a href="/wiki/Filter_(mathematics)" title="Filter (mathematics)">Filter (mathematics)</a></div> An ideal of the Boolean algebra A is a nonempty subset I such that for all x, y in I we have <var style="padding-right: 1px;">x</var> ∨ <var style="padding-right: 1px;">y</var> in I and for all a in A we have <var style="padding-right: 1px;">a</var> ∧ <var style="padding-right: 1px;">x</var> in I. This notion of ideal coincides with the notion of <a href="/wiki/Ring_ideal" class="mw-redirect" title="Ring ideal">ring ideal</a> in the Boolean ring A. An ideal I of A is called prime if <var style="padding-right: 1px;">I</var> ≠ <var style="padding-right: 1px;">A</var> and if <var style="padding-right: 1px;">a</var> ∧ <var style="padding-right: 1px;">b</var> in I always implies a in I or b in I. Furthermore, for every <var style="padding-right: 1px;">a</var> ∈ <var style="padding-right: 1px;">A</var> we have that <var style="padding-right: 1px;">a</var> ∧ −<var style="padding-right: 1px;">a</var> = 0 ∈ <var style="padding-right: 1px;">I</var>, and then if I is prime we have <var style="padding-right: 1px;">a</var> ∈ <var style="padding-right: 1px;">I</var> or −<var style="padding-right: 1px;">a</var> ∈ <var style="padding-right: 1px;">I</var> for every <var style="padding-right: 1px;">a</var> ∈ <var style="padding-right: 1px;">A</var>. An ideal I of A is called maximal if <var style="padding-right: 1px;">I</var> ≠ <var style="padding-right: 1px;">A</var> and if the only ideal properly containing I is A itself. For an ideal I, if <var style="padding-right: 1px;">a</var> ∉ <var style="padding-right: 1px;">I</var> and −<var style="padding-right: 1px;">a</var> ∉ <var style="padding-right: 1px;">I</var>, then <var style="padding-right: 1px;">I</var> ∪ {<var style="padding-right: 1px;">a</var>} or <var style="padding-right: 1px;">I</var> ∪ {−<var style="padding-right: 1px;">a</var>} is contained in another proper ideal J. Hence, such an I is not maximal, and therefore the notions of prime ideal and maximal ideal are equivalent in Boolean algebras. Moreover, these notions coincide with ring theoretic ones of <a href="/wiki/Prime_ideal" title="Prime ideal">prime ideal</a> and <a href="/wiki/Maximal_ideal" title="Maximal ideal">maximal ideal</a> in the Boolean ring A. The dual of an ideal is a filter. A filter of the Boolean algebra A is a nonempty subset p such that for all x, y in p we have <var style="padding-right: 1px;">x</var> ∧ <var style="padding-right: 1px;">y</var> in p and for all a in A we have <var style="padding-right: 1px;">a</var> ∨ <var style="padding-right: 1px;">x</var> in p. The dual of a maximal (or prime) ideal in a Boolean algebra is <a href="/wiki/Ultrafilter" title="Ultrafilter">ultrafilter</a>. Ultrafilters can alternatively be described as <a href="/wiki/2-valued_morphism" title="2-valued morphism">2-valued morphisms</a> from A to the two-element Boolean algebra. The statement every filter in a Boolean algebra can be extended to an ultrafilter is called the <a href="/wiki/Boolean_prime_ideal_theorem#The_ultrafilter_lemma" title="Boolean prime ideal theorem">ultrafilter lemma</a> and cannot be proven in <a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel set theory</a> (ZF), if <a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">ZF</a> is <a href="/wiki/Consistent" class="mw-redirect" title="Consistent">consistent</a>. Within ZF, the ultrafilter lemma is strictly weaker than the <a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a>. The ultrafilter lemma has many equivalent formulations: every Boolean algebra has an ultrafilter, every ideal in a Boolean algebra can be extended to a prime ideal, etc. <div class="mw-heading mw-heading2"><h2 id="Representations">Representations</h2>[<a href="/w/index.php?title=Boolean_algebra_(structure)&action=edit&section=7" title="Edit section: Representations">edit</a>]</div> It can be shown that every finite Boolean algebra is isomorphic to the Boolean algebra of all subsets of a finite set. Therefore, the number of elements of every finite Boolean algebra is a <a href="/wiki/Power_of_two" title="Power of two">power of two</a>. <a href="/wiki/Marshall_H._Stone" title="Marshall H. Stone">Stone's</a> celebrated <a href="/wiki/Stone%27s_representation_theorem_for_Boolean_algebras" title="Stone's representation theorem for Boolean algebras">representation theorem for Boolean algebras</a> states that every Boolean algebra A is isomorphic to the Boolean algebra of all <a href="/wiki/Clopen_set" title="Clopen set">clopen sets</a> in some (<a href="/wiki/Compact_space" title="Compact space">compact</a> <a href="/wiki/Totally_disconnected" class="mw-redirect" title="Totally disconnected">totally disconnected</a> <a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff</a>) topological space. <div class="mw-heading mw-heading2"><h2 id="Axiomatics">Axiomatics</h2>[<a href="/w/index.php?title=Boolean_algebra_(structure)&action=edit&section=8" title="Edit section: Axiomatics">edit</a>]</div> <table align="right" class="wikitable collapsible collapsed" style="text-align:left"> <tbody><tr> <th colspan="2">Proven properties </th></tr> <tr valign="top"> <td> <table align="left" class="collapsible collapsed" style="text-align:left"> <tbody><tr> <th>UId1</th> <th></th> <th colspan="2">If x ∨ o = x for all x, then o = 0 </th></tr> <tr> <td>Proof:</td> <td></td> <td colspan="2">If x ∨ o = x, then </td></tr> <tr> <td></td> <td></td> <td>0 </td></tr> <tr> <td></td> <td>=</td> <td>0 ∨ o</td> <td>by assumption </td></tr> <tr> <td></td> <td>=</td> <td>o ∨ 0</td> <td>by Cmm1 </td></tr> <tr> <td></td> <td>=</td> <td>o</td> <td>by Idn1 </td></tr></tbody></table> </td> <td>UId2   [dual]   If x ∧ i = x for all x, then i = 1 </td></tr> <tr valign="top"> <td> <table align="left" class="collapsible collapsed" style="text-align:left"> <tbody><tr> <th>Idm1</th> <th></th> <th>x ∨ x = x </th></tr> <tr> <td>Proof:</td> <td></td> <td>x ∨ x </td></tr> <tr> <td></td> <td>=</td> <td>(x ∨ x) ∧ 1</td> <td>by Idn2 </td></tr> <tr> <td></td> <td>=</td> <td>(x ∨ x) ∧ (x ∨ ¬x)</td> <td>by Cpl1 </td></tr> <tr> <td></td> <td>=</td> <td>x ∨ (x ∧ ¬x)</td> <td>by Dst1 </td></tr> <tr> <td></td> <td>=</td> <td>x ∨ 0</td> <td>by Cpl2 </td></tr> <tr> <td></td> <td>=</td> <td>x</td> <td>by Idn1 </td></tr></tbody></table> </td> <td>Idm2   [dual]   x ∧ x = x </td></tr> <tr valign="top"> <td> <table align="left" class="collapsible collapsed" style="text-align:left"> <tbody><tr> <th>Bnd1</th> <th></th> <th>x ∨ 1 = 1 </th></tr> <tr> <td>Proof:</td> <td></td> <td>x ∨ 1 </td></tr> <tr> <td></td> <td>=</td> <td>(x ∨ 1) ∧ 1</td> <td>by Idn2 </td></tr> <tr> <td></td> <td>=</td> <td>1 ∧ (x ∨ 1)</td> <td>by Cmm2 </td></tr> <tr> <td></td> <td>=</td> <td>(x ∨ ¬x) ∧ (x ∨ 1)</td> <td>by Cpl1 </td></tr> <tr> <td></td> <td>=</td> <td>x ∨ (¬x ∧ 1)</td> <td>by Dst1 </td></tr> <tr> <td></td> <td>=</td> <td>x ∨ ¬x</td> <td>by Idn2 </td></tr> <tr> <td></td> <td>=</td> <td>1</td> <td>by Cpl1 </td></tr></tbody></table> </td> <td>Bnd2   [dual]   x ∧ 0 = 0 </td></tr> <tr valign="top"> <td> <table align="left" class="collapsible collapsed" style="text-align:left"> <tbody><tr> <th>Abs1</th> <th></th> <th>x ∨ (x ∧ y) = x </th></tr> <tr> <td>Proof:</td> <td></td> <td>x ∨ (x ∧ y) </td></tr> <tr> <td></td> <td>=</td> <td>(x ∧ 1) ∨ (x ∧ y)</td> <td>by Idn2 </td></tr> <tr> <td></td> <td>=</td> <td>x ∧ (1 ∨ y)</td> <td>by Dst2 </td></tr> <tr> <td></td> <td>=</td> <td>x ∧ (y ∨ 1)</td> <td>by Cmm1 </td></tr> <tr> <td></td> <td>=</td> <td>x ∧ 1</td> <td>by Bnd1 </td></tr> <tr> <td></td> <td>=</td> <td>x</td> <td>by Idn2 </td></tr></tbody></table> </td> <td>Abs2   [dual]   x ∧ (x ∨ y) = x </td></tr> <tr valign="top"> <td colspan="2"> <table align="left" class="collapsible collapsed" style="text-align:left"> <tbody><tr> <th>UNg</th> <th></th> <th colspan="2">If x ∨ xn = 1 and x ∧ xn = 0, then xn = ¬x </th></tr> <tr> <td>Proof:</td> <td></td> <td colspan="2">If x ∨ xn = 1 and x ∧ xn = 0, then </td></tr> <tr> <td></td> <td></td> <td>xn </td></tr> <tr> <td></td> <td>=</td> <td>xn ∧ 1</td> <td>by Idn2 </td></tr> <tr> <td></td> <td>=</td> <td>xn ∧ (x ∨ ¬x)</td> <td>by Cpl1 </td></tr> <tr> <td></td> <td>=</td> <td>(xn ∧ x) ∨ (xn ∧ ¬x)</td> <td>by Dst2 </td></tr> <tr> <td></td> <td>=</td> <td>(x ∧ xn) ∨ (¬x ∧ xn)</td> <td>by Cmm2 </td></tr> <tr> <td></td> <td>=</td> <td>0 ∨ (¬x ∧ xn)</td> <td>by assumption </td></tr> <tr> <td></td> <td>=</td> <td>(x ∧ ¬x) ∨ (¬x ∧ xn)</td> <td>by Cpl2 </td></tr> <tr> <td></td> <td>=</td> <td>(¬x ∧ x) ∨ (¬x ∧ xn)</td> <td>by Cmm2 </td></tr> <tr> <td></td> <td>=</td> <td>¬x ∧ (x ∨ xn)</td> <td>by Dst2 </td></tr> <tr> <td></td> <td>=</td> <td>¬x ∧ 1</td> <td>by assumption </td></tr> <tr> <td></td> <td>=</td> <td>¬x</td> <td>by Idn2 </td></tr></tbody></table> </td></tr> <tr valign="top"> <td colspan="2"> <table align="left" class="collapsible collapsed" style="text-align:left"> <tbody><tr> <th>DNg</th> <th></th> <th>¬¬x = x </th></tr> <tr> <td>Proof:</td> <td></td> <td>¬x ∨ x = x ∨ ¬x = 1</td> <td>by Cmm1, Cpl1 </td></tr> <tr> <td></td> <td>and</td> <td>¬x ∧ x = x ∧ ¬x = 0</td> <td>by Cmm2, Cpl2 </td></tr> <tr> <td></td> <td>hence</td> <td>x = ¬¬x</td> <td>by UNg </td></tr></tbody></table> </td></tr> <tr valign="top"> <td> <table align="left" class="collapsible collapsed" style="text-align:left"> <tbody><tr> <th>A1</th> <th></th> <th>x ∨ (¬x ∨ y) = 1 </th></tr> <tr> <td>Proof:</td> <td></td> <td>x ∨ (¬x ∨ y) </td></tr> <tr> <td></td> <td>=</td> <td>(x ∨ (¬x ∨ y)) ∧ 1</td> <td>by Idn2 </td></tr> <tr> <td></td> <td>=</td> <td>1 ∧ (x ∨ (¬x ∨ y))</td> <td>by Cmm2 </td></tr> <tr> <td></td> <td>=</td> <td>(x ∨ ¬x) ∧ (x ∨ (¬x ∨ y))</td> <td>by Cpl1 </td></tr> <tr> <td></td> <td>=</td> <td>x ∨ (¬x ∧ (¬x ∨ y))</td> <td>by Dst1 </td></tr> <tr> <td></td> <td>=</td> <td>x ∨ ¬x</td> <td>by Abs2 </td></tr> <tr> <td></td> <td>=</td> <td>1</td> <td>by Cpl1 </td></tr></tbody></table> </td> <td>A2   [dual]   x ∧ (¬x ∧ y) = 0 </td></tr> <tr valign="top"> <td> <table align="left" class="collapsible collapsed" style="text-align:left"> <tbody><tr> <th>B1</th> <th></th> <th>(x ∨ y) ∨ (¬x ∧ ¬y) = 1 </th></tr> <tr> <td>Proof:</td> <td></td> <td>(x ∨ y) ∨ (¬x ∧ ¬y) </td></tr> <tr> <td></td> <td>=</td> <td>((x ∨ y) ∨ ¬x) ∧ ((x ∨ y) ∨ ¬y)</td> <td>by Dst1 </td></tr> <tr> <td></td> <td>=</td> <td>(¬x ∨ (x ∨ y)) ∧ (¬y ∨ (y ∨ x))</td> <td>by Cmm1 </td></tr> <tr> <td></td> <td>=</td> <td>(¬x ∨ (¬¬x ∨ y)) ∧ (¬y ∨ (¬¬y ∨ x))</td> <td>by DNg </td></tr> <tr> <td></td> <td>=</td> <td>1 ∧ 1</td> <td>by A1 </td></tr> <tr> <td></td> <td>=</td> <td>1</td> <td>by Idn2 </td></tr></tbody></table> </td> <td>B2   [dual]   (x ∧ y) ∧ (¬x ∨ ¬y) = 0 </td></tr> <tr valign="top"> <td> <table align="left" class="collapsible collapsed" style="text-align:left"> <tbody><tr> <th>C1</th> <th></th> <th>(x ∨ y) ∧ (¬x ∧ ¬y) = 0 </th></tr> <tr> <td>Proof:</td> <td></td> <td>(x ∨ y) ∧ (¬x ∧ ¬y) </td></tr> <tr> <td></td> <td>=</td> <td>(¬x ∧ ¬y) ∧ (x ∨ y)</td> <td>by Cmm2 </td></tr> <tr> <td></td> <td>=</td> <td>((¬x ∧ ¬y) ∧ x) ∨ ((¬x ∧ ¬y) ∧ y)</td> <td>by Dst2 </td></tr> <tr> <td></td> <td>=</td> <td>(x ∧ (¬x ∧ ¬y)) ∨ (y ∧ (¬y ∧ ¬x))</td> <td>by Cmm2 </td></tr> <tr> <td></td> <td>=</td> <td>0 ∨ 0</td> <td>by A2 </td></tr> <tr> <td></td> <td>=</td> <td>0</td> <td>by Idn1 </td></tr></tbody></table> </td> <td>C2   [dual]   (x ∧ y) ∨ (¬x ∨ ¬y) = 1 </td></tr> <tr valign="top"> <td> <table align="left" class="collapsible collapsed" style="text-align:left"> <tbody><tr> <th>DMg1</th> <th></th> <th>¬(x ∨ y) = ¬x ∧ ¬y </th></tr> <tr> <td>Proof:</td> <td></td> <td>by B1, C1, and UNg </td></tr></tbody></table> </td> <td>DMg2   [dual]   ¬(x ∧ y) = ¬x ∨ ¬y </td></tr> <tr valign="top"> <td> <table align="left" class="collapsible collapsed" style="text-align:left"> <tbody><tr> <th>D1</th> <th></th> <th>(x∨(y∨z)) ∨ ¬x = 1 </th></tr> <tr> <td>Proof:</td> <td></td> <td>(x ∨ (y ∨ z)) ∨ ¬x </td></tr> <tr> <td></td> <td>=</td> <td>¬x ∨ (x ∨ (y ∨ z))</td> <td>by Cmm1 </td></tr> <tr> <td></td> <td>=</td> <td>¬x ∨ (¬¬x ∨ (y ∨ z))</td> <td>by DNg </td></tr> <tr> <td></td> <td>=</td> <td>1</td> <td>by A1 </td></tr></tbody></table> </td> <td>D2   [dual]   (x∧(y∧z)) ∧ ¬x = 0 </td></tr> <tr valign="top"> <td> <table align="left" class="collapsible collapsed" style="text-align:left"> <tbody><tr> <th>E1</th> <th></th> <th>y ∧ (x∨(y∨z)) = y </th></tr> <tr> <td>Proof:</td> <td></td> <td>y ∧ (x ∨ (y ∨ z)) </td></tr> <tr> <td></td> <td>=</td> <td>(y ∧ x) ∨ (y ∧ (y ∨ z))</td> <td>by Dst2 </td></tr> <tr> <td></td> <td>=</td> <td>(y ∧ x) ∨ y</td> <td>by Abs2 </td></tr> <tr> <td></td> <td>=</td> <td>y ∨ (y ∧ x)</td> <td>by Cmm1 </td></tr> <tr> <td></td> <td>=</td> <td>y</td> <td>by Abs1 </td></tr></tbody></table> </td> <td>E2   [dual]   y ∨ (x∧(y∧z)) = y </td></tr> <tr valign="top"> <td> <table align="left" class="collapsible collapsed" style="text-align:left"> <tbody><tr> <th>F1</th> <th></th> <th>(x∨(y∨z)) ∨ ¬y = 1 </th></tr> <tr> <td>Proof:</td> <td></td> <td>(x ∨ (y ∨ z)) ∨ ¬y </td></tr> <tr> <td></td> <td>=</td> <td>¬y ∨ (x ∨ (y ∨ z))</td> <td>by Cmm1 </td></tr> <tr> <td></td> <td>=</td> <td>(¬y ∨ (x ∨ (y ∨ z))) ∧ 1</td> <td>by Idn2 </td></tr> <tr> <td></td> <td>=</td> <td>1 ∧ (¬y ∨ (x ∨ (y ∨ z)))</td> <td>by Cmm2 </td></tr> <tr> <td></td> <td>=</td> <td>(y ∨ ¬y) ∧ (¬y ∨ (x ∨ (y ∨ z)))</td> <td>by Cpl1 </td></tr> <tr> <td></td> <td>=</td> <td>(¬y ∨ y) ∧ (¬y ∨ (x ∨ (y ∨ z)))</td> <td>by Cmm1 </td></tr> <tr> <td></td> <td>=</td> <td>¬y ∨ (y ∧ (x ∨ (y ∨ z)))</td> <td>by Dst1 </td></tr> <tr> <td></td> <td>=</td> <td>¬y ∨ y</td> <td>by E1 </td></tr> <tr> <td></td> <td>=</td> <td>y ∨ ¬y</td> <td>by Cmm1 </td></tr> <tr> <td></td> <td>=</td> <td>1</td> <td>by Cpl1 </td></tr></tbody></table> </td> <td>F2   [dual]   (x∧(y∧z)) ∧ ¬y = 0 </td></tr> <tr valign="top"> <td> <table align="left" class="collapsible collapsed" style="text-align:left"> <tbody><tr> <th>G1</th> <th></th> <th>(x∨(y∨z)) ∨ ¬z = 1 </th></tr> <tr> <td>Proof:</td> <td></td> <td>(x ∨ (y ∨ z)) ∨ ¬z </td></tr> <tr> <td></td> <td>=</td> <td>(x ∨ (z ∨ y)) ∨ ¬z</td> <td>by Cmm1 </td></tr> <tr> <td></td> <td>=</td> <td>1</td> <td>by F1 </td></tr></tbody></table> </td> <td>G2   [dual]   (x∧(y∧z)) ∧ ¬z = 0 </td></tr> <tr valign="top"> <td> <table align="left" class="collapsible collapsed" style="text-align:left"> <tbody><tr> <th>H1</th> <th></th> <th>¬((x∨y)∨z) ∧ x = 0 </th></tr> <tr> <td>Proof:</td> <td></td> <td>¬((x ∨ y) ∨ z) ∧ x </td></tr> <tr> <td></td> <td>=</td> <td>(¬(x ∨ y) ∧ ¬z) ∧ x</td> <td>by DMg1 </td></tr> <tr> <td></td> <td>=</td> <td>((¬x ∧ ¬y) ∧ ¬z) ∧ x</td> <td>by DMg1 </td></tr> <tr> <td></td> <td>=</td> <td>x ∧ ((¬x ∧ ¬y) ∧ ¬z)</td> <td>by Cmm2 </td></tr> <tr> <td></td> <td>=</td> <td>(x ∧ ((¬x ∧ ¬y) ∧ ¬z)) ∨ 0</td> <td>by Idn1 </td></tr> <tr> <td></td> <td>=</td> <td>0 ∨ (x ∧ ((¬x ∧ ¬y) ∧ ¬z))</td> <td>by Cmm1 </td></tr> <tr> <td></td> <td>=</td> <td>(x ∧ ¬x) ∨ (x ∧ ((¬x ∧ ¬y) ∧ ¬z))</td> <td>by Cpl2 </td></tr> <tr> <td></td> <td>=</td> <td>x ∧ (¬x ∨ ((¬x ∧ ¬y) ∧ ¬z))</td> <td>by Dst2 </td></tr> <tr> <td></td> <td>=</td> <td>x ∧ (¬x ∨ (¬z ∧ (¬x ∧ ¬y)))</td> <td>by Cmm2 </td></tr> <tr> <td></td> <td>=</td> <td>x ∧ ¬x</td> <td>by E2 </td></tr> <tr> <td></td> <td>=</td> <td>0</td> <td>by Cpl2 </td></tr></tbody></table> </td> <td>H2   [dual]   ¬((x∧y)∧z) ∨ x = 1 </td></tr> <tr valign="top"> <td> <table align="left" class="collapsible collapsed" style="text-align:left"> <tbody><tr> <th>I1</th> <th></th> <th>¬((x∨y)∨z) ∧ y = 0 </th></tr> <tr> <td>Proof:</td> <td></td> <td>¬((x ∨ y) ∨ z) ∧ y </td></tr> <tr> <td></td> <td>=</td> <td>¬((y ∨ x) ∨ z) ∧ y</td> <td>by Cmm1 </td></tr> <tr> <td></td> <td>=</td> <td>0</td> <td>by H1 </td></tr></tbody></table> </td> <td>I2   [dual]   ¬((x∧y)∧z) ∨ y = 1 </td></tr> <tr valign="top"> <td> <table align="left" class="collapsible collapsed" style="text-align:left"> <tbody><tr> <th>J1</th> <th></th> <th>¬((x∨y)∨z) ∧ z = 0 </th></tr> <tr> <td>Proof:</td> <td></td> <td>¬((x ∨ y) ∨ z) ∧ z </td></tr> <tr> <td></td> <td>=</td> <td>(¬(x ∨ y) ∧ ¬z) ∧ z</td> <td>by DMg1 </td></tr> <tr> <td></td> <td>=</td> <td>z ∧ (¬(x ∨ y) ∧ ¬z)</td> <td>by Cmm2 </td></tr> <tr> <td></td> <td>=</td> <td>z ∧ ( ¬z ∧ ¬(x ∨ y))</td> <td>by Cmm2 </td></tr> <tr> <td></td> <td>=</td> <td>0</td> <td>by A2 </td></tr></tbody></table> </td> <td>J2   [dual]   ¬((x∧y)∧z) ∨ z = 1 </td></tr> <tr valign="top"> <td> <table align="left" class="collapsible collapsed" style="text-align:left"> <tbody><tr> <th>K1</th> <th></th> <th>(x ∨ (y ∨ z)) ∨ ¬((x ∨ y) ∨ z) = 1 </th></tr> <tr> <td>Proof:</td> <td></td> <td>(x∨(y∨z)) ∨ ¬((x ∨ y) ∨ z) </td></tr> <tr> <td></td> <td>=</td> <td>(x∨(y∨z)) ∨ (¬(x ∨ y) ∧ ¬z)</td> <td>by DMg1 </td></tr> <tr> <td></td> <td>=</td> <td>(x∨(y∨z)) ∨ ((¬x ∧ ¬y) ∧ ¬z)</td> <td>by DMg1 </td></tr> <tr> <td></td> <td>=</td> <td>((x∨(y∨z)) ∨ (¬x ∧ ¬y)) ∧ ((x∨(y∨z)) ∨ ¬z)</td> <td>by Dst1 </td></tr> <tr> <td></td> <td>=</td> <td>(((x∨(y∨z)) ∨ ¬x) ∧ ((x∨(y∨z)) ∨ ¬y)) ∧ ((x∨(y∨z)) ∨ ¬z)</td> <td>by Dst1 </td></tr> <tr> <td></td> <td>=</td> <td>(1 ∧ 1) ∧ 1</td> <td>by D1,F1,G1 </td></tr> <tr> <td></td> <td>=</td> <td>1</td> <td>by Idn2 </td></tr></tbody></table> </td> <td>K2   [dual]   (x ∧ (y ∧ z)) ∧ ¬((x ∧ y) ∧ z) = 0 </td></tr> <tr valign="top"> <td> <table align="left" class="collapsible collapsed" style="text-align:left"> <tbody><tr> <th>L1</th> <th></th> <th>(x ∨ (y ∨ z)) ∧ ¬((x ∨ y) ∨ z) = 0 </th></tr> <tr> <td>Proof:</td> <td></td> <td>(x ∨ (y ∨ z)) ∧ ¬((x ∨ y) ∨ z) </td></tr> <tr> <td></td> <td>=</td> <td>¬((x∨y)∨z) ∧ (x ∨ (y ∨ z))</td> <td>by Cmm2 </td></tr> <tr> <td></td> <td>=</td> <td>(¬((x∨y)∨z) ∧ x) ∨ (¬((x∨y)∨z) ∧ (y ∨ z))</td> <td>by Dst2 </td></tr> <tr> <td></td> <td>=</td> <td>(¬((x∨y)∨z) ∧ x) ∨ ((¬((x∨y)∨z) ∧ y) ∨ (¬((x∨y)∨z) ∧ z))</td> <td>by Dst2 </td></tr> <tr> <td></td> <td>=</td> <td>0 ∨ (0 ∨ 0)</td> <td>by H1,I1,J1 </td></tr> <tr> <td></td> <td>=</td> <td>0</td> <td>by Idn1 </td></tr></tbody></table> </td> <td>L2   [dual]   (x ∧ (y ∧ z)) ∨ ¬((x ∧ y) ∧ z) = 1 </td></tr> <tr valign="top"> <td> <table align="left" class="collapsible collapsed" style="text-align:left"> <tbody><tr> <th>Ass1</th> <th></th> <th>x ∨ (y ∨ z) = (x ∨ y) ∨ z </th></tr> <tr> <td>Proof:</td> <td></td> <td>by K1, L1, UNg, DNg </td></tr></tbody></table> </td> <td>Ass2   [dual]   x ∧ (y ∧ z) = (x ∧ y) ∧ z </td></tr> <tr> <td colspan="2"> <table align="left" class="collapsible" style="text-align:left"> <tbody><tr> <th colspan="2">Abbreviations </th></tr> <tr> <td>UId</td> <td>Unique Identity </td></tr> <tr> <td>Idm</td> <td><a href="/wiki/Idempotence" title="Idempotence">Idempotence</a> </td></tr> <tr> <td>Bnd</td> <td><a href="/wiki/Bounded_lattice" class="mw-redirect" title="Bounded lattice">Boundaries</a> </td></tr> <tr> <td>Abs</td> <td><a href="/wiki/Absorption_law" title="Absorption law">Absorption law</a> </td></tr> <tr> <td>UNg</td> <td>Unique Negation </td></tr> <tr> <td>DNg</td> <td><a href="/wiki/Double_negation" title="Double negation">Double negation</a> </td></tr> <tr> <td>DMg</td> <td><a href="/wiki/De_Morgan%27s_Law" class="mw-redirect" title="De Morgan's Law">De Morgan's Law</a> </td></tr> <tr> <td>Ass</td> <td><a href="/wiki/Associativity" class="mw-redirect" title="Associativity">Associativity</a> </td></tr></tbody></table> </td></tr></tbody></table> <table align="right" class="wikitable collapsible collapsed" style="text-align:left"> <tbody><tr> <th colspan="4">Huntington 1904 Boolean algebra axioms </th></tr> <tr valign="top"> <td>Idn1</td> <td>x ∨ 0 = x </td> <td>Idn2</td> <td>x ∧ 1 = x </td></tr> <tr valign="top"> <td>Cmm1</td> <td>x ∨ y = y ∨ x </td> <td>Cmm2</td> <td>x ∧ y = y ∧ x </td></tr> <tr valign="top"> <td>Dst1</td> <td>x ∨ (y∧z) = (x∨y) ∧ (x∨z) </td> <td>Dst2</td> <td>x ∧ (y∨z) = (x∧y) ∨ (x∧z) </td></tr> <tr valign="top"> <td>Cpl1</td> <td>x ∨ ¬x = 1 </td> <td>Cpl2</td> <td>x ∧ ¬x = 0 </td></tr> <tr> <td colspan="4"> <table align="left" class="collapsible" style="text-align:left"> <tbody><tr> <th colspan="2">Abbreviations </th></tr> <tr> <td>Idn</td> <td><a href="/wiki/Identity_element" title="Identity element">Identity</a> </td></tr> <tr> <td>Cmm</td> <td><a href="/wiki/Commutativity" class="mw-redirect" title="Commutativity">Commutativity</a> </td></tr> <tr> <td>Dst</td> <td><a href="/wiki/Distributivity" class="mw-redirect" title="Distributivity">Distributivity</a> </td></tr> <tr> <td>Cpl</td> <td><a href="/wiki/Complemented_lattice" title="Complemented lattice">Complements</a> </td></tr></tbody></table> </td></tr></tbody></table> The first axiomatization of Boolean lattices/algebras in general was given by the English philosopher and mathematician <a href="/wiki/Alfred_North_Whitehead" title="Alfred North Whitehead">Alfred North Whitehead</a> in 1898.<a href="#cite_note-FOOTNOTEPadmanabhanRudeanu2008[httpsbooksgooglecombooksidJlXSlpmlSv4CpgPA73_73]-8">[7]</a><a href="#cite_note-FOOTNOTEWhitehead189837-9">[8]</a> It included the <a href="#Definition">above axioms</a> and additionally x ∨ 1 = 1 and x ∧ 0 = 0. In 1904, the American mathematician <a href="/wiki/Edward_V._Huntington" class="mw-redirect" title="Edward V. Huntington">Edward V. Huntington</a> (1874–1952) gave probably the most parsimonious axiomatization based on ∧, ∨, ¬, even proving the associativity laws (see box).<a href="#cite_note-FOOTNOTEHuntington1904292–293-10">[9]</a> He also proved that these axioms are <a href="/wiki/Independence_(mathematical_logic)" title="Independence (mathematical logic)">independent</a> of each other.<a href="#cite_note-FOOTNOTEHuntington1904296-11">[10]</a> In 1933, Huntington set out the following elegant axiomatization for Boolean algebra.<a href="#cite_note-FOOTNOTEHuntington1933a-12">[11]</a> It requires just one binary operation + and a <a href="/wiki/Unary_functional_symbol" class="mw-redirect" title="Unary functional symbol">unary functional symbol</a> n, to be read as 'complement', which satisfy the following laws: <div><ol><li>Commutativity: x + y = y + x.</li><li>Associativity: (x + y) + z = x + (y + z).</li><li>Huntington equation: n(n(x) + y) + n(n(x) + n(y)) = x.</li></ol></div> <a href="/wiki/Herbert_Robbins" title="Herbert Robbins">Herbert Robbins</a> immediately asked: If the Huntington equation is replaced with its dual, to wit: <div><ol start="4"><li>Robbins Equation: n(n(x + y) + n(x + n(y))) = x,</li></ol></div> do (1), (2), and (4) form a basis for Boolean algebra? Calling (1), (2), and (4) a Robbins algebra, the question then becomes: Is every Robbins algebra a Boolean algebra? This question (which came to be known as the <a href="/wiki/Robbins_conjecture" class="mw-redirect" title="Robbins conjecture">Robbins conjecture</a>) remained open for decades, and became a favorite question of <a href="/wiki/Alfred_Tarski" title="Alfred Tarski">Alfred Tarski</a> and his students. In 1996, <a href="/wiki/William_McCune" title="William McCune">William McCune</a> at <a href="/wiki/Argonne_National_Laboratory" title="Argonne National Laboratory">Argonne National Laboratory</a>, building on earlier work by Larry Wos, Steve Winker, and Bob Veroff, answered Robbins's question in the affirmative: Every Robbins algebra is a Boolean algebra. Crucial to McCune's proof was the computer program <a href="/wiki/Equational_prover" title="Equational prover">EQP</a> he designed. For a simplification of McCune's proof, see Dahn (1998). Further work has been done for reducing the number of axioms; see <a href="/wiki/Minimal_axioms_for_Boolean_algebra" title="Minimal axioms for Boolean algebra">Minimal axioms for Boolean algebra</a>. <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2>[<a href="/w/index.php?title=Boolean_algebra_(structure)&action=edit&section=9" title="Edit section: Generalizations">edit</a>]</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 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(mathematics)">Group</a></li> <li><a href="/wiki/Semigroup" title="Semigroup">Semigroup</a> / <a href="/wiki/Monoid" title="Monoid">Monoid</a></li> <li><a href="/wiki/Racks_and_quandles" title="Racks and quandles">Rack and quandle</a></li> <li><a href="/wiki/Quasigroup" title="Quasigroup">Quasigroup and loop</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Abelian_group" title="Abelian group">Abelian group</a></li> <li><a href="/wiki/Magma_(algebra)" title="Magma (algebra)">Magma</a></li> <li><a href="/wiki/Lie_group" title="Lie group">Lie group</a></li></ul> </div> <a href="/wiki/Group_theory" title="Group theory">Group theory</a></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/Ring_(mathematics)" title="Ring (mathematics)">Ring</a>-like</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">Ring</a></li> <li><a href="/wiki/Rng_(algebra)" title="Rng (algebra)">Rng</a></li> <li><a href="/wiki/Semiring" title="Semiring">Semiring</a></li> <li><a href="/wiki/Near-ring" title="Near-ring">Near-ring</a></li> <li><a href="/wiki/Commutative_ring" title="Commutative ring">Commutative ring</a></li> <li><a href="/wiki/Domain_(ring_theory)" title="Domain (ring theory)">Domain</a></li> <li><a href="/wiki/Integral_domain" title="Integral domain">Integral domain</a></li> <li><a href="/wiki/Field_(mathematics)" title="Field (mathematics)">Field</a></li> <li><a href="/wiki/Division_ring" title="Division ring">Division ring</a></li> <li><a href="/wiki/Lie_algebra#Lie_ring" title="Lie algebra">Lie ring</a></li></ul> </div> <a href="/wiki/Ring_theory" title="Ring theory">Ring theory</a></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/Lattice_(order)" title="Lattice (order)">Lattice</a>-like</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Lattice_(order)" title="Lattice (order)">Lattice</a></li> <li><a href="/wiki/Semilattice" title="Semilattice">Semilattice</a></li> <li><a href="/wiki/Complemented_lattice" title="Complemented lattice">Complemented lattice</a></li> <li><a href="/wiki/Total_order" title="Total order">Total order</a></li> <li><a href="/wiki/Heyting_algebra" title="Heyting algebra">Heyting algebra</a></li> <li><a class="mw-selflink selflink">Boolean algebra</a></li></ul> </div> <ul><li><a href="/wiki/Map_of_lattices" title="Map of lattices">Map of lattices</a></li> <li><a href="/wiki/Lattice_(order)" title="Lattice (order)">Lattice theory</a></li></ul></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/Module_(mathematics)" title="Module (mathematics)">Module</a>-like</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Module_(mathematics)" title="Module (mathematics)">Module</a></li> <li><a href="/wiki/Group_with_operators" title="Group with operators">Group with operators</a></li> <li><a href="/wiki/Vector_space" title="Vector space">Vector space</a></li></ul> </div> <ul><li><a href="/wiki/Linear_algebra" title="Linear algebra">Linear algebra</a></li></ul></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/Algebra_over_a_field" title="Algebra over a field">Algebra</a>-like</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Algebra_over_a_field" title="Algebra over a field">Algebra</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Associative_algebra" title="Associative algebra">Associative</a></li> <li><a href="/wiki/Non-associative_algebra" title="Non-associative algebra">Non-associative</a></li> <li><a href="/wiki/Composition_algebra" title="Composition algebra">Composition algebra</a></li> <li><a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a></li> <li><a href="/wiki/Graded_ring" title="Graded ring">Graded</a></li> <li><a href="/wiki/Bialgebra" title="Bialgebra">Bialgebra</a></li> <li><a href="/wiki/Hopf_algebra" title="Hopf algebra">Hopf algebra</a></li></ul> </div></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:Algebraic_structures" title="Template:Algebraic structures"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Algebraic_structures" title="Template talk:Algebraic structures"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Algebraic_structures" title="Special:EditPage/Template:Algebraic structures"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> Removing the requirement of existence of a unit from the axioms of Boolean algebra yields "generalized Boolean algebras". Formally, a <a href="/wiki/Distributive_lattice" title="Distributive lattice">distributive lattice</a> B is a generalized Boolean lattice, if it has a smallest element 0 and for any elements a and b in B such that a ≤ b, there exists an element x such that a ∧ x = 0 and a ∨ x = b. Defining a \ b as the unique x such that (a ∧ b) ∨ x = a and (a ∧ b) ∧ x = 0, we say that the structure (B, ∧, ∨, \, 0) is a generalized Boolean algebra, while (B, ∨, 0) is a generalized Boolean <a href="/wiki/Semilattice" title="Semilattice">semilattice</a>. Generalized Boolean lattices are exactly the <a href="/wiki/Ideal_(order_theory)" title="Ideal (order theory)">ideals</a> of Boolean lattices. A structure that satisfies all axioms for Boolean algebras except the two distributivity axioms is called an <a href="/wiki/Orthocomplemented_lattice" class="mw-redirect" title="Orthocomplemented lattice">orthocomplemented lattice</a>. Orthocomplemented lattices arise naturally in <a href="/wiki/Quantum_logic" title="Quantum logic">quantum logic</a> as lattices of <a href="/wiki/Closed_set" title="Closed set">closed</a> <a href="/wiki/Linear_subspace" title="Linear subspace">linear subspaces</a> for <a href="/wiki/Separable_space" title="Separable space">separable</a> <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert spaces</a>. <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Boolean_algebra_(structure)&action=edit&section=10" title="Edit section: See also">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col"> <ul><li><a href="/wiki/List_of_Boolean_algebra_topics" title="List of Boolean algebra topics">List of Boolean algebra topics</a></li> <li><a href="/wiki/Boolean_domain" title="Boolean domain">Boolean domain</a></li> <li><a href="/wiki/Boolean_function" title="Boolean function">Boolean function</a></li> <li><a href="/wiki/Boolean_logic" class="mw-redirect" title="Boolean logic">Boolean logic</a></li> <li><a href="/wiki/Boolean_ring" title="Boolean ring">Boolean ring</a></li> <li><a href="/wiki/Boolean-valued_function" title="Boolean-valued function">Boolean-valued function</a></li> <li><a href="/wiki/Canonical_form_(Boolean_algebra)" class="mw-redirect" title="Canonical form (Boolean algebra)">Canonical form (Boolean algebra)</a></li> <li><a href="/wiki/Complete_Boolean_algebra" title="Complete Boolean algebra">Complete Boolean algebra</a></li> <li><a href="/wiki/De_Morgan%27s_laws" title="De Morgan's laws">De Morgan's laws</a></li> <li><a href="/wiki/Forcing_(mathematics)" title="Forcing (mathematics)">Forcing (mathematics)</a></li> <li><a href="/wiki/Free_Boolean_algebra" title="Free Boolean algebra">Free Boolean algebra</a></li> <li><a href="/wiki/Heyting_algebra" title="Heyting algebra">Heyting algebra</a></li> <li><a href="/wiki/Hypercube_graph" title="Hypercube graph">Hypercube graph</a></li> <li><a href="/wiki/Karnaugh_map" title="Karnaugh map">Karnaugh map</a></li> <li><a href="/wiki/Laws_of_Form" title="Laws of Form">Laws of Form</a></li> <li><a href="/wiki/Logic_gate" title="Logic gate">Logic gate</a></li> <li><a href="/wiki/Logical_graph" class="mw-redirect" title="Logical graph">Logical graph</a></li> <li><a href="/wiki/Logical_matrix" title="Logical matrix">Logical matrix</a></li> <li><a href="/wiki/Propositional_logic" class="mw-redirect" title="Propositional logic">Propositional logic</a></li> <li><a href="/wiki/Quine%E2%80%93McCluskey_algorithm" title="Quine–McCluskey algorithm">Quine–McCluskey algorithm</a></li> <li><a href="/wiki/Two-element_Boolean_algebra" title="Two-element Boolean algebra">Two-element Boolean algebra</a></li> <li><a href="/wiki/Venn_diagram" title="Venn diagram">Venn diagram</a></li> <li><a href="/wiki/Conditional_event_algebra" title="Conditional event algebra">Conditional event algebra</a></li></ul></div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Boolean_algebra_(structure)&action=edit&section=11" 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"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-4"><a href="#cite_ref-4">^</a> Strictly, electrical engineers tend to use additional states to represent other circuit conditions such as high impedance - see <a href="/wiki/IEEE_1164" title="IEEE 1164">IEEE 1164</a> or <a href="/wiki/IEEE_1364" class="mw-redirect" title="IEEE 1364">IEEE 1364</a>. </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Boolean_algebra_(structure)&action=edit&section=12" title="Edit section: References">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-FOOTNOTEGivantHalmos200920-1"><a href="#cite_ref-FOOTNOTEGivantHalmos200920_1-0">^</a> <a href="#CITEREFGivantHalmos2009">Givant & Halmos 2009</a>, p. 20. </li> <li id="cite_note-FOOTNOTEDaveyPriestley1990109,_131,_144-2"><a href="#cite_ref-FOOTNOTEDaveyPriestley1990109,_131,_144_2-0">^</a> <a href="#CITEREFDaveyPriestley1990">Davey & Priestley 1990</a>, pp. 109, 131, 144. </li> <li id="cite_note-FOOTNOTEGoodstein201221ff-3"><a href="#cite_ref-FOOTNOTEGoodstein201221ff_3-0">^</a> <a href="#CITEREFGoodstein2012">Goodstein 2012</a>, p. 21ff. </li> <li id="cite_note-FOOTNOTEStone1936-5"><a href="#cite_ref-FOOTNOTEStone1936_5-0">^</a> <a href="#CITEREFStone1936">Stone 1936</a>. </li> <li id="cite_note-FOOTNOTEHsiang1985260-6"><a href="#cite_ref-FOOTNOTEHsiang1985260_6-0">^</a> <a href="#CITEREFHsiang1985">Hsiang 1985</a>, p. 260. </li> <li id="cite_note-FOOTNOTECohn2003[httpsbooksgooglecombooksidVESm0MJOiDQCpgPA81_81]-7"><a href="#cite_ref-FOOTNOTECohn2003[httpsbooksgooglecombooksidVESm0MJOiDQCpgPA81_81]_7-0">^</a> <a href="#CITEREFCohn2003">Cohn 2003</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=VESm0MJOiDQC&pg=PA81">81</a>. </li> <li id="cite_note-FOOTNOTEPadmanabhanRudeanu2008[httpsbooksgooglecombooksidJlXSlpmlSv4CpgPA73_73]-8"><a href="#cite_ref-FOOTNOTEPadmanabhanRudeanu2008[httpsbooksgooglecombooksidJlXSlpmlSv4CpgPA73_73]_8-0">^</a> <a href="#CITEREFPadmanabhanRudeanu2008">Padmanabhan & Rudeanu 2008</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=JlXSlpmlSv4C&pg=PA73">73</a>. </li> <li id="cite_note-FOOTNOTEWhitehead189837-9"><a href="#cite_ref-FOOTNOTEWhitehead189837_9-0">^</a> <a href="#CITEREFWhitehead1898">Whitehead 1898</a>, p. 37. </li> <li id="cite_note-FOOTNOTEHuntington1904292–293-10"><a href="#cite_ref-FOOTNOTEHuntington1904292–293_10-0">^</a> <a href="#CITEREFHuntington1904">Huntington 1904</a>, pp. 292–293. </li> <li id="cite_note-FOOTNOTEHuntington1904296-11"><a href="#cite_ref-FOOTNOTEHuntington1904296_11-0">^</a> <a href="#CITEREFHuntington1904">Huntington 1904</a>, p. 296. </li> <li id="cite_note-FOOTNOTEHuntington1933a-12"><a href="#cite_ref-FOOTNOTEHuntington1933a_12-0">^</a> <a href="#CITEREFHuntington1933a">Huntington 1933a</a>. </li> </ol></div> <div class="mw-heading mw-heading3"><h3 id="Works_cited">Works cited</h3>[<a href="/w/index.php?title=Boolean_algebra_(structure)&action=edit&section=13" title="Edit section: Works cited">edit</a>]</div> <ul><li><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="CITEREFDaveyPriestley1990" class="citation book cs1">Davey, B.A.; <a href="/wiki/Hilary_Priestley" title="Hilary Priestley">Priestley, H.A.</a> (1990). <a href="/wiki/Introduction_to_Lattices_and_Order" title="Introduction to Lattices and Order">Introduction to Lattices and Order</a>. Cambridge Mathematical Textbooks. Cambridge University Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Lattices+and+Order&rft.series=Cambridge+Mathematical+Textbooks&rft.pub=Cambridge+University+Press&rft.date=1990&rft.aulast=Davey&rft.aufirst=B.A.&rft.au=Priestley%2C+H.A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABoolean+algebra+%28structure%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCohn2003" class="citation cs2"><a href="/wiki/Paul_Cohn" title="Paul Cohn">Cohn, Paul M.</a> (2003), Basic Algebra: Groups, Rings, and Fields, Springer, pp. 51, 70–81, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781852335878" title="Special:BookSources/9781852335878"><bdi>9781852335878</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Basic+Algebra%3A+Groups%2C+Rings%2C+and+Fields&rft.pages=51%2C+70-81&rft.pub=Springer&rft.date=2003&rft.isbn=9781852335878&rft.aulast=Cohn&rft.aufirst=Paul+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABoolean+algebra+%28structure%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGivantHalmos2009" class="citation cs2">Givant, Steven; <a href="/wiki/Paul_Halmos" title="Paul Halmos">Halmos, Paul</a> (2009), Introduction to Boolean Algebras, <a href="/wiki/Undergraduate_Texts_in_Mathematics" title="Undergraduate Texts in Mathematics">Undergraduate Texts in Mathematics</a>, <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-40293-2" title="Special:BookSources/978-0-387-40293-2"><bdi>978-0-387-40293-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Boolean+Algebras&rft.series=Undergraduate+Texts+in+Mathematics&rft.pub=Springer&rft.date=2009&rft.isbn=978-0-387-40293-2&rft.aulast=Givant&rft.aufirst=Steven&rft.au=Halmos%2C+Paul&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABoolean+algebra+%28structure%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoodstein2012" class="citation cs2"><a href="/wiki/Reuben_Goodstein" title="Reuben Goodstein">Goodstein, R. L.</a> (2012), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=0fxW2KiyxWwC&pg=PA21">"Chapter 2: The self-dual system of axioms"</a>, Boolean Algebra, Courier Dover Publications, pp. 21ff, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780486154978" title="Special:BookSources/9780486154978"><bdi>9780486154978</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+2%3A+The+self-dual+system+of+axioms&rft.btitle=Boolean+Algebra&rft.pages=21ff&rft.pub=Courier+Dover+Publications&rft.date=2012&rft.isbn=9780486154978&rft.aulast=Goodstein&rft.aufirst=R.+L.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D0fxW2KiyxWwC%26pg%3DPA21&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABoolean+algebra+%28structure%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHsiang1985" class="citation journal cs1">Hsiang, Jieh (1985). <a rel="nofollow" class="external text" href="https://www.researchgate.net/publication/223327412">"Refutational Theorem Proving Using Term Rewriting Systems"</a>. <a href="/wiki/Artificial_Intelligence_(journal)" title="Artificial Intelligence (journal)">Artificial Intelligence</a>. 25 (3): 255–300. <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%2F0004-3702%2885%2990074-8">10.1016/0004-3702(85)90074-8</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Artificial+Intelligence&rft.atitle=Refutational+Theorem+Proving+Using+Term+Rewriting+Systems&rft.volume=25&rft.issue=3&rft.pages=255-300&rft.date=1985&rft_id=info%3Adoi%2F10.1016%2F0004-3702%2885%2990074-8&rft.aulast=Hsiang&rft.aufirst=Jieh&rft_id=https%3A%2F%2Fwww.researchgate.net%2Fpublication%2F223327412&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABoolean+algebra+%28structure%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHuntington1904" class="citation journal cs1"><a href="/wiki/Edward_V._Huntington" class="mw-redirect" title="Edward V. Huntington">Huntington, Edward V.</a> (1904). <a rel="nofollow" class="external text" href="https://zenodo.org/record/1431563">"Sets of Independent Postulates for the Algebra of Logic"</a>. <a href="/wiki/Transactions_of_the_American_Mathematical_Society" title="Transactions of the American Mathematical Society">Transactions of the American Mathematical Society</a>. 5 (3): 288–309. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fs0002-9947-1904-1500675-4">10.1090/s0002-9947-1904-1500675-4</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1986459">1986459</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transactions+of+the+American+Mathematical+Society&rft.atitle=Sets+of+Independent+Postulates+for+the+Algebra+of+Logic&rft.volume=5&rft.issue=3&rft.pages=288-309&rft.date=1904&rft_id=info%3Adoi%2F10.1090%2Fs0002-9947-1904-1500675-4&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1986459%23id-name%3DJSTOR&rft.aulast=Huntington&rft.aufirst=Edward+V.&rft_id=https%3A%2F%2Fzenodo.org%2Frecord%2F1431563&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABoolean+algebra+%28structure%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPadmanabhanRudeanu2008" class="citation cs2">Padmanabhan, Ranganathan; Rudeanu, Sergiu (2008), Axioms for lattices and boolean algebras, World Scientific, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-283-454-6" title="Special:BookSources/978-981-283-454-6"><bdi>978-981-283-454-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Axioms+for+lattices+and+boolean+algebras&rft.pub=World+Scientific&rft.date=2008&rft.isbn=978-981-283-454-6&rft.aulast=Padmanabhan&rft.aufirst=Ranganathan&rft.au=Rudeanu%2C+Sergiu&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABoolean+algebra+%28structure%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStone1936" class="citation journal cs1"><a href="/wiki/Marshall_H._Stone" title="Marshall H. Stone">Stone, Marshall H.</a> (1936). <a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fs0002-9947-1936-1501865-8">"The Theory of Representations for Boolean Algebra"</a>. <a href="/wiki/Transactions_of_the_American_Mathematical_Society" title="Transactions of the American Mathematical Society">Transactions of the American Mathematical Society</a>. 40: 37–111. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fs0002-9947-1936-1501865-8">10.1090/s0002-9947-1936-1501865-8</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transactions+of+the+American+Mathematical+Society&rft.atitle=The+Theory+of+Representations+for+Boolean+Algebra&rft.volume=40&rft.pages=37-111&rft.date=1936&rft_id=info%3Adoi%2F10.1090%2Fs0002-9947-1936-1501865-8&rft.aulast=Stone&rft.aufirst=Marshall+H.&rft_id=https%3A%2F%2Fdoi.org%2F10.1090%252Fs0002-9947-1936-1501865-8&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABoolean+algebra+%28structure%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWhitehead1898" class="citation book cs1"><a href="/wiki/A.N._Whitehead" class="mw-redirect" title="A.N. Whitehead">Whitehead, A.N.</a> (1898). <a rel="nofollow" class="external text" href="http://projecteuclid.org/euclid.chmm/1263316509">A Treatise on Universal Algebra</a>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4297-0032-0" title="Special:BookSources/978-1-4297-0032-0"><bdi>978-1-4297-0032-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Treatise+on+Universal+Algebra&rft.pub=Cambridge+University+Press&rft.date=1898&rft.isbn=978-1-4297-0032-0&rft.aulast=Whitehead&rft.aufirst=A.N.&rft_id=http%3A%2F%2Fprojecteuclid.org%2Feuclid.chmm%2F1263316509&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABoolean+algebra+%28structure%29" class="Z3988"></li></ul> <div class="mw-heading mw-heading3"><h3 id="General_references">General references</h3>[<a href="/w/index.php?title=Boolean_algebra_(structure)&action=edit&section=14" title="Edit section: General references">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_footnotes_needed plainlinks metadata ambox ambox-style ambox-More_footnotes_needed" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/40px-Text_document_with_red_question_mark.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/60px-Text_document_with_red_question_mark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/80px-Text_document_with_red_question_mark.svg.png 2x" data-file-width="48" data-file-height="48" /></div></td><td class="mbox-text"><div class="mbox-text-span">This article includes a list of <a href="/wiki/Wikipedia:Citing_sources#General_references" title="Wikipedia:Citing sources">general references</a>, but it lacks sufficient corresponding <a href="/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">inline citations</a>. Please help to <a href="/wiki/Wikipedia:WikiProject_Reliability" title="Wikipedia:WikiProject Reliability">improve</a> this article by <a href="/wiki/Wikipedia:When_to_cite" title="Wikipedia:When to cite">introducing</a> more precise citations. (July 2013) (<a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a>)</div></td></tr></tbody></table> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrownVranesic2002" class="citation cs2">Brown, Stephen; Vranesic, Zvonko (2002), Fundamentals of Digital Logic with VHDL Design (2nd ed.), <a href="/wiki/McGraw-Hill" class="mw-redirect" title="McGraw-Hill">McGraw–Hill</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-249938-4" title="Special:BookSources/978-0-07-249938-4"><bdi>978-0-07-249938-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+Digital+Logic+with+VHDL+Design&rft.edition=2nd&rft.pub=McGraw%E2%80%93Hill&rft.date=2002&rft.isbn=978-0-07-249938-4&rft.aulast=Brown&rft.aufirst=Stephen&rft.au=Vranesic%2C+Zvonko&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABoolean+algebra+%28structure%29" class="Z3988">. See Section 2.5.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBoudetJouannaudSchmidt-Schauß1989" class="citation journal cs1">Boudet, A.; Jouannaud, J.P.; Schmidt-Schauß, M. (1989). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fs0747-7171%2889%2980054-9">"Unification in Boolean Rings and Abelian Groups"</a>. <a href="/wiki/Journal_of_Symbolic_Computation" title="Journal of Symbolic Computation">Journal of Symbolic Computation</a>. 8 (5): 449–477. <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%2Fs0747-7171%2889%2980054-9">10.1016/s0747-7171(89)80054-9</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Symbolic+Computation&rft.atitle=Unification+in+Boolean+Rings+and+Abelian+Groups&rft.volume=8&rft.issue=5&rft.pages=449-477&rft.date=1989&rft_id=info%3Adoi%2F10.1016%2Fs0747-7171%2889%2980054-9&rft.aulast=Boudet&rft.aufirst=A.&rft.au=Jouannaud%2C+J.P.&rft.au=Schmidt-Schau%C3%9F%2C+M.&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252Fs0747-7171%252889%252980054-9&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABoolean+algebra+%28structure%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCoriLascar2000" class="citation cs2">Cori, Rene; Lascar, Daniel (2000), Mathematical Logic: A Course with Exercises, <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-850048-3" title="Special:BookSources/978-0-19-850048-3"><bdi>978-0-19-850048-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Logic%3A+A+Course+with+Exercises&rft.pub=Oxford+University+Press&rft.date=2000&rft.isbn=978-0-19-850048-3&rft.aulast=Cori&rft.aufirst=Rene&rft.au=Lascar%2C+Daniel&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABoolean+algebra+%28structure%29" class="Z3988">. See Chapter 2.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDahn1998" class="citation cs2">Dahn, B. I. (1998), "Robbins Algebras are Boolean: A Revision of McCune's Computer-Generated Solution of the Robbins Problem", <a href="/wiki/Journal_of_Algebra" title="Journal of Algebra">Journal of Algebra</a>, 208 (2): 526–532, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Fjabr.1998.7467">10.1006/jabr.1998.7467</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Algebra&rft.atitle=Robbins+Algebras+are+Boolean%3A+A+Revision+of+McCune%27s+Computer-Generated+Solution+of+the+Robbins+Problem&rft.volume=208&rft.issue=2&rft.pages=526-532&rft.date=1998&rft_id=info%3Adoi%2F10.1006%2Fjabr.1998.7467&rft.aulast=Dahn&rft.aufirst=B.+I.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABoolean+algebra+%28structure%29" class="Z3988">.</li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHalmos1963" class="citation cs2"><a href="/wiki/Paul_Halmos" title="Paul Halmos">Halmos, Paul</a> (1963), Lectures on Boolean Algebras, Van Nostrand, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-90094-0" title="Special:BookSources/978-0-387-90094-0"><bdi>978-0-387-90094-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lectures+on+Boolean+Algebras&rft.pub=Van+Nostrand&rft.date=1963&rft.isbn=978-0-387-90094-0&rft.aulast=Halmos&rft.aufirst=Paul&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABoolean+algebra+%28structure%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHalmosGivant1998" class="citation cs2"><a href="/wiki/Paul_Halmos" title="Paul Halmos">Halmos, Paul</a>; Givant, Steven (1998), <a rel="nofollow" class="external text" href="https://archive.org/details/logicasalgebra0000halm">Logic as Algebra</a>, Dolciani Mathematical Expositions, vol. 21, <a href="/wiki/Mathematical_Association_of_America" title="Mathematical Association of America">Mathematical Association of America</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-88385-327-6" title="Special:BookSources/978-0-88385-327-6"><bdi>978-0-88385-327-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Logic+as+Algebra&rft.series=Dolciani+Mathematical+Expositions&rft.pub=Mathematical+Association+of+America&rft.date=1998&rft.isbn=978-0-88385-327-6&rft.aulast=Halmos&rft.aufirst=Paul&rft.au=Givant%2C+Steven&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Flogicasalgebra0000halm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABoolean+algebra+%28structure%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHuntington1933a" class="citation cs2"><a href="/wiki/Edward_Vermilye_Huntington" title="Edward Vermilye Huntington">Huntington, E. V.</a> (1933a), <a rel="nofollow" class="external text" href="https://www.ams.org/journals/tran/1933-035-01/S0002-9947-1933-1501684-X/S0002-9947-1933-1501684-X.pdf">"New sets of independent postulates for the algebra of logic"</a> (PDF), <a href="/wiki/Transactions_of_the_American_Mathematical_Society" title="Transactions of the American Mathematical Society">Transactions of the American Mathematical Society</a>, 35 (1), American Mathematical Society: 274–304, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1989325">10.2307/1989325</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1989325">1989325</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transactions+of+the+American+Mathematical+Society&rft.atitle=New+sets+of+independent+postulates+for+the+algebra+of+logic&rft.volume=35&rft.issue=1&rft.pages=274-304&rft.date=1933&rft_id=info%3Adoi%2F10.2307%2F1989325&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1989325%23id-name%3DJSTOR&rft.aulast=Huntington&rft.aufirst=E.+V.&rft_id=https%3A%2F%2Fwww.ams.org%2Fjournals%2Ftran%2F1933-035-01%2FS0002-9947-1933-1501684-X%2FS0002-9947-1933-1501684-X.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABoolean+algebra+%28structure%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHuntington1933b" class="citation cs2"><a href="/wiki/Edward_Vermilye_Huntington" title="Edward Vermilye Huntington">Huntington, E. V.</a> (1933b), "Boolean algebra: A correction", <a href="/wiki/Transactions_of_the_American_Mathematical_Society" title="Transactions of the American Mathematical Society">Transactions of the American Mathematical Society</a>, 35 (2): 557–558, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1989783">10.2307/1989783</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1989783">1989783</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transactions+of+the+American+Mathematical+Society&rft.atitle=Boolean+algebra%3A+A+correction&rft.volume=35&rft.issue=2&rft.pages=557-558&rft.date=1933&rft_id=info%3Adoi%2F10.2307%2F1989783&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1989783%23id-name%3DJSTOR&rft.aulast=Huntington&rft.aufirst=E.+V.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABoolean+algebra+%28structure%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMendelson1970" class="citation cs2">Mendelson, Elliott (1970), <a rel="nofollow" class="external text" href="https://archive.org/details/schaumsoutlineof00mend">Boolean Algebra and Switching Circuits</a>, Schaum's Outline Series in Mathematics, <a href="/wiki/McGraw-Hill" class="mw-redirect" title="McGraw-Hill">McGraw–Hill</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-041460-0" title="Special:BookSources/978-0-07-041460-0"><bdi>978-0-07-041460-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Boolean+Algebra+and+Switching+Circuits&rft.series=Schaum%27s+Outline+Series+in+Mathematics&rft.pub=McGraw%E2%80%93Hill&rft.date=1970&rft.isbn=978-0-07-041460-0&rft.aulast=Mendelson&rft.aufirst=Elliott&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fschaumsoutlineof00mend&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABoolean+algebra+%28structure%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMonkBonnet1989" class="citation cs2">Monk, J. Donald; Bonnet, R., eds. (1989), <a rel="nofollow" class="external text" href="https://archive.org/details/handbookofboolea0000unse">Handbook of Boolean Algebras</a>, <a href="/wiki/Elsevier" title="Elsevier">North-Holland</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-444-87291-3" title="Special:BookSources/978-0-444-87291-3"><bdi>978-0-444-87291-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+Boolean+Algebras&rft.pub=North-Holland&rft.date=1989&rft.isbn=978-0-444-87291-3&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fhandbookofboolea0000unse&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABoolean+algebra+%28structure%29" class="Z3988">. In 3 volumes. (Vol.1:<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/978-0-444-70261-6" title="Special:BookSources/978-0-444-70261-6">978-0-444-70261-6</a>, Vol.2:<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/978-0-444-87152-7" title="Special:BookSources/978-0-444-87152-7">978-0-444-87152-7</a>, Vol.3:<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/978-0-444-87153-4" title="Special:BookSources/978-0-444-87153-4">978-0-444-87153-4</a>)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="Sikorski1966BooleanAlgebras" class="citation cs2"><a href="/wiki/Roman_Sikorski" title="Roman Sikorski">Sikorski, Roman</a> (1966), Boolean Algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, <a href="/wiki/Springer_Verlag" class="mw-redirect" title="Springer Verlag">Springer Verlag</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Boolean+Algebras&rft.series=Ergebnisse+der+Mathematik+und+ihrer+Grenzgebiete&rft.pub=Springer+Verlag&rft.date=1966&rft.aulast=Sikorski&rft.aufirst=Roman&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABoolean+algebra+%28structure%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStoll1963" class="citation cs2">Stoll, R. R. (1963), Set Theory and Logic, W. H. Freeman, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-63829-4" title="Special:BookSources/978-0-486-63829-4"><bdi>978-0-486-63829-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Set+Theory+and+Logic&rft.pub=W.+H.+Freeman&rft.date=1963&rft.isbn=978-0-486-63829-4&rft.aulast=Stoll&rft.aufirst=R.+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABoolean+algebra+%28structure%29" class="Z3988">. Reprinted by <a href="/wiki/Dover_Publications" title="Dover Publications">Dover Publications</a>, 1979.</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Boolean_algebra_(structure)&action=edit&section=15" title="Edit section: External links">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-External_links plainlinks metadata ambox ambox-style ambox-external_links" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/40px-Edit-clear.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/60px-Edit-clear.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/80px-Edit-clear.svg.png 2x" data-file-width="48" data-file-height="48" /></div></td><td class="mbox-text"><div class="mbox-text-span">This article's use of <a href="/wiki/Wikipedia:External_links" title="Wikipedia:External links">external links</a> may not follow Wikipedia's policies or guidelines. 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(November 2020) (<a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a>)</div></td></tr></tbody></table> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Boolean_algebra">"Boolean algebra"</a>, <a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Boolean+algebra&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DBoolean_algebra&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABoolean+algebra+%28structure%29" class="Z3988"></li> <li><a href="/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a>: "<a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/boolalg-math/">The Mathematics of Boolean Algebra</a>", by J. Donald Monk.</li> <li>McCune W., 1997. <a rel="nofollow" class="external text" href="http://www.cs.unm.edu/~mccune/papers/robbins/">Robbins Algebras Are Boolean</a> JAR 19(3), 263–276</li> <li><a rel="nofollow" class="external text" href="http://demonstrations.wolfram.com/BooleanAlgebra/">"Boolean Algebra"</a> by <a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Eric W. Weisstein</a>, <a href="/wiki/Wolfram_Demonstrations_Project" title="Wolfram Demonstrations Project">Wolfram Demonstrations Project</a>, 2007.</li> <li>Burris, Stanley N.; Sankappanavar, H. P., 1981. <a rel="nofollow" class="external text" href="http://www.thoralf.uwaterloo.ca/htdocs/ualg.html">A Course in Universal Algebra.</a> Springer-Verlag. <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/3-540-90578-2" title="Special:BookSources/3-540-90578-2">3-540-90578-2</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/BooleanAlgebra.html">"Boolean Algebra"</a>. <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Boolean+Algebra&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FBooleanAlgebra.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABoolean+algebra+%28structure%29" class="Z3988"></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output 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class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/List_of_order_theory_topics" title="List of order theory topics">Topics</a></li> <li><a href="/wiki/Glossary_of_order_theory" title="Glossary of order theory">Glossary</a></li> <li><a href="/wiki/Category:Order_theory" title="Category:Order theory">Category</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Key concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Binary_relation" title="Binary relation">Binary relation</a></li> <li><a class="mw-selflink selflink">Boolean algebra</a></li> <li><a href="/wiki/Cyclic_order" title="Cyclic order">Cyclic order</a></li> <li><a href="/wiki/Lattice_(order)" title="Lattice (order)">Lattice</a></li> <li><a href="/wiki/Partially_ordered_set" title="Partially ordered set">Partial order</a></li> <li><a href="/wiki/Preorder" title="Preorder">Preorder</a></li> <li><a href="/wiki/Total_order" title="Total order">Total order</a></li> <li><a href="/wiki/Weak_ordering" title="Weak ordering">Weak ordering</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Results</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Boolean_prime_ideal_theorem" title="Boolean prime ideal theorem">Boolean prime ideal theorem</a></li> <li><a href="/wiki/Cantor%E2%80%93Bernstein_theorem" title="Cantor–Bernstein theorem">Cantor–Bernstein theorem</a></li> <li><a href="/wiki/Cantor%27s_isomorphism_theorem" title="Cantor's isomorphism theorem">Cantor's isomorphism theorem</a></li> <li><a href="/wiki/Dilworth%27s_theorem" title="Dilworth's theorem">Dilworth's theorem</a></li> <li><a href="/wiki/Dushnik%E2%80%93Miller_theorem" title="Dushnik–Miller theorem">Dushnik–Miller theorem</a></li> <li><a href="/wiki/Hausdorff_maximal_principle" title="Hausdorff maximal principle">Hausdorff maximal principle</a></li> <li><a href="/wiki/Knaster%E2%80%93Tarski_theorem" title="Knaster–Tarski theorem">Knaster–Tarski theorem</a></li> <li><a href="/wiki/Kruskal%27s_tree_theorem" title="Kruskal's tree theorem">Kruskal's tree theorem</a></li> <li><a href="/wiki/Laver%27s_theorem" title="Laver's theorem">Laver's theorem</a></li> <li><a href="/wiki/Mirsky%27s_theorem" title="Mirsky's theorem">Mirsky's theorem</a></li> <li><a href="/wiki/Szpilrajn_extension_theorem" title="Szpilrajn extension theorem">Szpilrajn extension theorem</a></li> <li><a href="/wiki/Zorn%27s_lemma" title="Zorn's lemma">Zorn's lemma</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties & Types (<a href="/wiki/List_of_order_structures_in_mathematics" title="List of order structures in mathematics">list</a>)</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Antisymmetric_relation" title="Antisymmetric relation">Antisymmetric</a></li> <li><a href="/wiki/Asymmetric_relation" title="Asymmetric relation">Asymmetric</a></li> <li><a class="mw-selflink selflink">Boolean algebra</a> <ul><li><a href="/wiki/List_of_Boolean_algebra_topics" title="List of Boolean algebra topics">topics</a></li></ul></li> <li><a href="/wiki/Completeness_(order_theory)" title="Completeness (order theory)">Completeness</a></li> <li><a href="/wiki/Connected_relation" title="Connected relation">Connected</a></li> <li><a href="/wiki/Covering_relation" title="Covering relation">Covering</a></li> <li><a href="/wiki/Dense_order" title="Dense order">Dense</a></li> <li><a href="/wiki/Directed_set" title="Directed set">Directed</a></li> <li>(<a href="/wiki/Partial_equivalence_relation" title="Partial equivalence relation">Partial</a>) <a href="/wiki/Equivalence_relation" title="Equivalence relation">Equivalence</a></li> <li><a href="/wiki/Foundational_relation" class="mw-redirect" title="Foundational relation">Foundational</a></li> <li><a href="/wiki/Heyting_algebra" title="Heyting algebra">Heyting algebra</a></li> <li><a href="/wiki/Homogeneous_relation" title="Homogeneous relation">Homogeneous</a></li> <li><a href="/wiki/Idempotent_relation" title="Idempotent relation">Idempotent</a></li> <li><a href="/wiki/Lattice_(order)" title="Lattice (order)">Lattice</a> <ul><li><a href="/wiki/Bounded_lattice" class="mw-redirect" title="Bounded lattice">Bounded</a></li> <li><a href="/wiki/Complemented_lattice" title="Complemented lattice">Complemented</a></li> <li><a href="/wiki/Complete_lattice" title="Complete lattice">Complete</a></li> <li><a href="/wiki/Distributive_lattice" title="Distributive lattice">Distributive</a></li> <li><a href="/wiki/Join_and_meet" title="Join and meet">Join and meet</a></li></ul></li> <li><a href="/wiki/Reflexive_relation" title="Reflexive relation">Reflexive</a></li> <li><a href="/wiki/Partial_order" class="mw-redirect" title="Partial order">Partial order</a> <ul><li><a href="/wiki/Chain-complete_partial_order" class="mw-redirect" title="Chain-complete partial order">Chain-complete</a></li> <li><a href="/wiki/Graded_poset" title="Graded poset">Graded</a></li> <li><a href="/wiki/Eulerian_poset" title="Eulerian poset">Eulerian</a></li> <li><a href="/wiki/Strict_partial_order" class="mw-redirect" title="Strict partial order">Strict</a></li></ul></li> <li><a href="/wiki/Prefix_order" title="Prefix order">Prefix order</a></li> <li><a href="/wiki/Preorder" title="Preorder">Preorder</a> <ul><li><a href="/wiki/Total_preorder" class="mw-redirect" title="Total preorder">Total</a></li></ul></li> <li><a href="/wiki/Semilattice" title="Semilattice">Semilattice</a></li> <li><a href="/wiki/Semiorder" title="Semiorder">Semiorder</a></li> <li><a href="/wiki/Symmetric_relation" title="Symmetric relation">Symmetric</a></li> <li><a href="/wiki/Total_relation" title="Total relation">Total</a></li> <li><a href="/wiki/Tolerance_relation" title="Tolerance relation">Tolerance</a></li> <li><a href="/wiki/Transitive_relation" title="Transitive relation">Transitive</a></li> <li><a href="/wiki/Well-founded_relation" title="Well-founded relation">Well-founded</a></li> <li><a href="/wiki/Well-quasi-ordering" title="Well-quasi-ordering">Well-quasi-ordering</a> (<a href="/wiki/Better-quasi-ordering" title="Better-quasi-ordering">Better</a>)</li> <li>(<a href="/wiki/Prewellordering" title="Prewellordering">Pre</a>) <a href="/wiki/Well-order" title="Well-order">Well-order</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Constructions</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Composition_of_relations" title="Composition of relations">Composition</a></li> <li><a href="/wiki/Converse_relation" title="Converse relation">Converse/Transpose</a></li> <li><a href="/wiki/Lexicographic_order" title="Lexicographic order">Lexicographic order</a></li> <li><a href="/wiki/Linear_extension" title="Linear extension">Linear extension</a></li> <li><a href="/wiki/Product_order" title="Product order">Product order</a></li> <li><a href="/wiki/Reflexive_closure" title="Reflexive closure">Reflexive closure</a></li> <li><a href="/wiki/Series-parallel_partial_order" title="Series-parallel partial order">Series-parallel partial order</a></li> <li><a href="/wiki/Star_product" title="Star product">Star product</a></li> <li><a href="/wiki/Symmetric_closure" title="Symmetric closure">Symmetric closure</a></li> <li><a href="/wiki/Transitive_closure" title="Transitive closure">Transitive closure</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Topology" title="Topology">Topology</a> & Orders</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alexandrov_topology" title="Alexandrov topology">Alexandrov topology</a> & <a href="/wiki/Specialization_(pre)order" title="Specialization (pre)order">Specialization preorder</a></li> <li><a href="/wiki/Ordered_topological_vector_space" title="Ordered topological vector space">Ordered topological vector space</a> <ul><li><a href="/wiki/Normal_cone_(functional_analysis)" title="Normal cone (functional analysis)">Normal cone</a></li> <li><a href="/wiki/Order_topology_(functional_analysis)" title="Order topology (functional analysis)">Order topology</a></li></ul></li> <li><a href="/wiki/Order_topology" title="Order topology">Order topology</a></li> <li><a href="/wiki/Topological_vector_lattice" title="Topological vector lattice">Topological vector lattice</a> <ul><li><a href="/wiki/Banach_lattice" title="Banach lattice">Banach</a></li> <li><a href="/wiki/Fr%C3%A9chet_lattice" title="Fréchet lattice">Fréchet</a></li> <li><a href="/wiki/Locally_convex_vector_lattice" title="Locally convex vector lattice">Locally convex</a></li> <li><a href="/wiki/Normed_lattice" class="mw-redirect" title="Normed lattice">Normed</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Antichain" title="Antichain">Antichain</a></li> <li><a href="/wiki/Cofinal_(mathematics)" title="Cofinal (mathematics)">Cofinal</a></li> <li><a href="/wiki/Cofinality" title="Cofinality">Cofinality</a></li> <li><a href="/wiki/Comparability" title="Comparability">Comparability</a> <ul><li><a href="/wiki/Comparability_graph" title="Comparability graph">Graph</a></li></ul></li> <li><a href="/wiki/Duality_(order_theory)" title="Duality (order theory)">Duality</a></li> <li><a href="/wiki/Filter_(mathematics)" title="Filter (mathematics)">Filter</a></li> <li><a href="/wiki/Hasse_diagram" title="Hasse diagram">Hasse diagram</a></li> <li><a href="/wiki/Ideal_(order_theory)" title="Ideal (order theory)">Ideal</a></li> <li><a href="/wiki/Net_(mathematics)" title="Net (mathematics)">Net</a> <ul><li><a href="/wiki/Subnet_(mathematics)" title="Subnet (mathematics)">Subnet</a></li></ul></li> <li><a href="/wiki/Monotonic_function" title="Monotonic function">Order morphism</a> <ul><li><a href="/wiki/Order_embedding" title="Order embedding">Embedding</a></li> <li><a href="/wiki/Order_isomorphism" title="Order isomorphism">Isomorphism</a></li></ul></li> <li><a href="/wiki/Order_type" title="Order type">Order type</a></li> <li><a href="/wiki/Ordered_field" title="Ordered field">Ordered field</a> <ul><li><a href="/wiki/Positive_cone_of_an_ordered_field" class="mw-redirect" title="Positive cone of an ordered field">Positive cone of an ordered field</a></li></ul></li> <li><a href="/wiki/Ordered_vector_space" title="Ordered vector space">Ordered vector space</a> <ul><li><a href="/wiki/Partially_ordered_space" title="Partially ordered space">Partially ordered</a></li> <li><a href="/wiki/Positive_cone_of_an_ordered_vector_space" class="mw-redirect" title="Positive cone of an ordered vector space">Positive cone of an ordered vector space</a></li> <li><a href="/wiki/Riesz_space" title="Riesz space">Riesz space</a></li></ul></li> <li><a href="/wiki/Partially_ordered_group" title="Partially ordered group">Partially ordered group</a> <ul><li><a href="/wiki/Positive_cone_of_a_partially_ordered_group" class="mw-redirect" title="Positive cone of a partially ordered group">Positive cone of a partially ordered group</a></li></ul></li> <li><a href="/wiki/Upper_set" title="Upper set">Upper set</a></li> <li><a href="/wiki/Young%27s_lattice" title="Young's lattice">Young's lattice</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img 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