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It does not yet have a dedicated thread; feel free to create one, giving it the same name as the title of this page" style="color:black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="algebra">Algebra</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/universal+algebra">universal algebra</a></p> <h2 id="algebraic_theories">Algebraic theories</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+theory">algebraic theory</a> / <a class="existingWikiWord" href="/nlab/show/2-algebraic+theory">2-algebraic theory</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-algebraic+theory">(∞,1)-algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monad">monad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-monad">(∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/operad">operad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-operad">(∞,1)-operad</a></p> </li> </ul> <h2 id="algebras_and_modules">Algebras and modules</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+a+monad">algebra over a monad</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-monad">∞-algebra over an (∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+algebraic+theory">algebra over an algebraic theory</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-algebraic+theory">∞-algebra over an (∞,1)-algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+operad">algebra over an operad</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-operad">∞-algebra over an (∞,1)-operad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action">action</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representation">representation</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/associated+bundle">associated bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a></p> </li> </ul> <h2 id="higher_algebras">Higher algebras</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+%28%E2%88%9E%2C1%29-category">monoidal (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid+in+an+%28%E2%88%9E%2C1%29-category">monoid in an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+an+%28%E2%88%9E%2C1%29-category">commutative monoid in an (∞,1)-category</a></p> </li> </ul> </li> <li> <p>symmetric monoidal (∞,1)-category of spectra</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smash+product+of+spectra">smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+smash+product+of+spectra">symmetric monoidal smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ring+spectrum">ring spectrum</a>, <a class="existingWikiWord" href="/nlab/show/module+spectrum">module spectrum</a>, <a class="existingWikiWord" href="/nlab/show/algebra+spectrum">algebra spectrum</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+algebra">A-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+ring">A-∞ ring</a>, <a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+space">A-∞ space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/C-%E2%88%9E+algebra">C-∞ algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a>, <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+algebra">E-∞ algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-module+bundle">(∞,1)-module bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multiplicative+cohomology+theory">multiplicative cohomology theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/deformation+theory">deformation theory</a></li> </ul> </li> </ul> <h2 id="model_category_presentations">Model category presentations</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+T-algebras">model structure on simplicial T-algebras</a> / <a class="existingWikiWord" href="/nlab/show/homotopy+T-algebra">homotopy T-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+operads">model structure on operads</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+an+operad">model structure on algebras over an operad</a></p> </li> </ul> <h2 id="geometry_on_formal_duals_of_algebras">Geometry on formal duals of algebras</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+geometry">derived geometry</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+conjecture">Deligne conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/delooping+hypothesis">delooping hypothesis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/higher+algebra+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="monoid_theory">Monoid theory</h4> <div class="hide"><div> <p><strong>monoid theory</strong> in <a class="existingWikiWord" href="/nlab/show/algebra">algebra</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid">monoid</a>, <a class="existingWikiWord" href="/nlab/show/infinity-monoid">infinity-monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid+object">monoid object</a>, <a class="existingWikiWord" href="/nlab/show/monoid+object+in+an+%28infinity%2C1%29-category">monoid object in an (infinity,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/semiring">semiring</a>, <a class="existingWikiWord" href="/nlab/show/rig">rig</a>, <a class="existingWikiWord" href="/nlab/show/ring">ring</a>, <a class="existingWikiWord" href="/nlab/show/associative+unital+algebra">associative unital algebra</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Mon">Mon</a>, <a class="existingWikiWord" href="/nlab/show/CMon">CMon</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid+homomorphism">monoid homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/trivial+monoid">trivial monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/submonoid">submonoid</a>, <span class="newWikiWord">quotient monoid<a href="/nlab/new/quotient+monoid">?</a></span></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/divisor">divisor</a>, <span class="newWikiWord">multiple<a href="/nlab/new/multiple">?</a></span>, <span class="newWikiWord">quotient element<a href="/nlab/new/quotient+element">?</a></span></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inverse+element">inverse element</a>, <a class="existingWikiWord" href="/nlab/show/unit">unit</a>, <a class="existingWikiWord" href="/nlab/show/irreducible+element">irreducible element</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ideal+in+a+monoid">ideal in a monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+ideal+in+a+monoid">principal ideal in a monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutative+monoid">commutative monoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/tensor+product+of+commutative+monoids">tensor product of commutative monoids</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cancellative+monoid">cancellative monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GCD+monoid">GCD monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unique+factorization+monoid">unique factorization monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/B%C3%A9zout+monoid">Bézout monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+ideal+monoid">principal ideal monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group">group</a>, <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/absorption+monoid">absorption monoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/zero+divisor">zero divisor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integral+monoid">integral monoid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/free+monoid">free monoid</a>, <a class="existingWikiWord" href="/nlab/show/free+commutative+monoid">free commutative monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/graphic+monoid">graphic monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid+action">monoid action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module+over+a+monoid">module over a monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/localization+of+a+monoid">localization of a monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+completion">group completion</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/endomorphism+monoid">endomorphism monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+commutative+monoid">super commutative monoid</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/monoid+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#definition'>Definition</a></li> <li><a href='#properties_and_constructions'>Properties and constructions</a></li> <ul> <li><a href='#ideals_forming_a_quantale'>Ideals forming a quantale</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> </ul> </div> <h2 id="definition">Definition</h2> <p>Given a <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> (or <a class="existingWikiWord" href="/nlab/show/semigroup">semigroup</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>, a <strong>left ideal</strong> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/subset">subset</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mi>A</mi></mrow><annotation encoding="application/x-tex">S A</annotation></semantics></math> is contained in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>. Similarly, a <strong>right ideal</strong> is a subset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mi>S</mi><mo>⊆</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">A S \subseteq A</annotation></semantics></math>. Finally, a <strong>two-sided ideal</strong>, or simply <strong>ideal</strong>, in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> is a subset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> that is both a left ideal and a right ideal.</p> <p>Given a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mi>I</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(C, \otimes, I)</annotation></semantics></math> and a <a class="existingWikiWord" href="/nlab/show/monoid+object">monoid object</a> (or semigroup object) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, we can <a class="existingWikiWord" href="/nlab/show/internalisation">internalise</a> the above. For instance, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>:</mo><mi>S</mi><mo>⊗</mo><mi>S</mi><mo>→</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">m: S \otimes S \to S</annotation></semantics></math> is the binary multiplication and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mo>=</mo><mi>m</mi><mo>∘</mo><mo stretchy="false">(</mo><mi>m</mi><mo>⊗</mo><msub><mn>1</mn> <mi>S</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>m</mi><mo>∘</mo><mo stretchy="false">(</mo><msub><mn>1</mn> <mi>S</mi></msub><mo>⊗</mo><mi>m</mi><mo stretchy="false">)</mo><mo>:</mo><mi>S</mi><mo>⊗</mo><mi>S</mi><mo>⊗</mo><mi>S</mi><mo>→</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">\mu = m \circ (m \otimes 1_S) = m \circ (1_S \otimes m): S \otimes S \otimes S \to S</annotation></semantics></math> the ternary multiplication, a two-sided ideal is a <a class="existingWikiWord" href="/nlab/show/subobject">subobject</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>, i.e., a mono <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">i: A \to S</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, such that the composite</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>⊗</mo><mi>A</mi><mo>⊗</mo><mi>S</mi><mover><mo>→</mo><mrow><msub><mn>1</mn> <mi>S</mi></msub><mo>⊗</mo><mi>i</mi><mo>⊗</mo><msub><mn>1</mn> <mi>S</mi></msub></mrow></mover><mi>S</mi><mo>⊗</mo><mi>S</mi><mo>⊗</mo><mi>S</mi><mover><mo>⟶</mo><mi>μ</mi></mover><mi>S</mi></mrow><annotation encoding="application/x-tex">S \otimes A \otimes S \stackrel{1_S \otimes i \otimes 1_S}{\to} S \otimes S \otimes S \stackrel{\mu}{\longrightarrow} S</annotation></semantics></math></div> <p>factors through <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">i: A \to S</annotation></semantics></math>. Clearly <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">i: A \to S</annotation></semantics></math> is not necessarily a <a class="existingWikiWord" href="/nlab/show/submonoid">submonoid</a>, inasmuch as the monoid unit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mo>:</mo><mi>I</mi><mo>→</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">e: I \to S</annotation></semantics></math> need not factor through <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">i: A \to S</annotation></semantics></math>.</p> <p>In particular for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">C = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a>, a monoid in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/ring">ring</a> and the corresponding notion of <em><a class="existingWikiWord" href="/nlab/show/ideal+in+a+ring">ideal in a ring</a></em> is the most common notion of ideal.</p> <p>See <a class="existingWikiWord" href="/nlab/show/ideal">ideal</a> for ideals in more well known contexts: commutative idempotent monoids (<a class="existingWikiWord" href="/nlab/show/semilattices">semilattices</a>) and monoids in <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a> (<a class="existingWikiWord" href="/nlab/show/rings">rings</a>).</p> <h2 id="properties_and_constructions">Properties and constructions</h2> <p>An ideal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> (on either side) must be a <span class="newWikiWord">subsemigroup<a href="/nlab/new/subsemigroup">?</a></span> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>, but it is a <a class="existingWikiWord" href="/nlab/show/submonoid">submonoid</a> iff <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">1 \in A</annotation></semantics></math>, in which case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">A = S</annotation></semantics></math>.</p> <h3 id="ideals_forming_a_quantale">Ideals forming a quantale</h3> <p>(Two-sided) ideals of a monoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> are frequently the elements of a <a class="existingWikiWord" href="/nlab/show/quantale">quantale</a> whose multiplication is called taking the <em>product of ideals</em>. In the classical case of ideals over a <a class="existingWikiWord" href="/nlab/show/ring">ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>, the product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mi>J</mi></mrow><annotation encoding="application/x-tex">I J</annotation></semantics></math> of ideals <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>,</mo><mi>J</mi><mo>⊆</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">I, J \subseteq R</annotation></semantics></math> is the smallest ideal containing all products <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mi>j</mi><mo>:</mo><mi>i</mi><mo>∈</mo><mi>I</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mi>J</mi></mrow><annotation encoding="application/x-tex">i j: i \in I, j \in J</annotation></semantics></math>; the sup-lattice of such ideals ordered by inclusion is a <a class="existingWikiWord" href="/nlab/show/residuated+lattice">residuated lattice</a>, in that there are also division operations where</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">/</mo><mi>J</mi><mo>≔</mo><mo stretchy="false">{</mo><mi>r</mi><mo>∈</mo><mi>R</mi><mo>:</mo><mi>r</mi><mi>J</mi><mo>⊆</mo><mi>K</mi><mo stretchy="false">}</mo><mo>;</mo><mspace width="2em"></mspace><mi>I</mi><mo>\</mo><mi>K</mi><mo>=</mo><mo stretchy="false">{</mo><mi>r</mi><mo>∈</mo><mi>R</mi><mo>:</mo><mi>I</mi><mi>r</mi><mo>⊆</mo><mi>K</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">K/J \coloneqq \{r \in R: r J \subseteq K\}; \qquad I\backslash K = \{r \in R: I r \subseteq K\}</annotation></semantics></math></div> <p>satisfying the expected <a class="existingWikiWord" href="/nlab/show/adjunction">adjointness</a> relations: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>⊆</mo><mi>K</mi><mo stretchy="false">/</mo><mi>J</mi></mrow><annotation encoding="application/x-tex">I \subseteq K/J</annotation></semantics></math> iff <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mi>J</mi><mo>⊆</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">I J \subseteq K</annotation></semantics></math> iff <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi><mo>⊆</mo><mi>I</mi><mo>\</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">J \subseteq I\backslash K</annotation></semantics></math>.</p> <p>A reasonably general context might be as follows.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>C</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{C}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/well-powered+category">well-powered</a> <a class="existingWikiWord" href="/nlab/show/regular+category">regular</a> <a class="existingWikiWord" href="/nlab/show/cosmos">cosmos</a> (‘cosmos’ in the sense of complete cocomplete symmetric monoidal closed category). Just using the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>C</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{C}</annotation></semantics></math> is a cosmos, we may construct a <a class="existingWikiWord" href="/nlab/show/monoidal+bicategory">monoidal bicategory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Mod</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>C</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Mod(\mathbf{C})</annotation></semantics></math> whose objects are monoids <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>C</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{C}</annotation></semantics></math>, whose 1-morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>→</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">S \to T</annotation></semantics></math> are left-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> right-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/modules">modules</a>, and whose 2-morphisms are bimodule homomorphisms.</p> <p>For each monoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>, there is a subbicategory of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Mod</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>C</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Mod(\mathbf{C})</annotation></semantics></math> whose only object is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>; this is a complete and cocomplete <a class="existingWikiWord" href="/nlab/show/biclosed+monoidal+category">biclosed monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>S</mi></msub></mrow><annotation encoding="application/x-tex">Mod_S</annotation></semantics></math> whose objects are bimodules, i.e., 1-morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>→</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">S \to S</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Mod</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>C</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Mod(\mathbf{C})</annotation></semantics></math>, and whose morphisms are bimodule homomorphisms. The unit of the monoidal product is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> with its standard <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-bimodule structure, and hence the <a class="existingWikiWord" href="/nlab/show/slice+category">slice</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>S</mi></msub><mo stretchy="false">/</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">Mod_S/S</annotation></semantics></math> (see also <a class="existingWikiWord" href="/nlab/show/semicartesian+monoidal+category">semicartesian monoidal category</a>) forms another complete and cocomplete biclosed monoidal category.</p> <p>An ideal of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> is just a subobject of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>S</mi></msub></mrow><annotation encoding="application/x-tex">Mod_S</annotation></semantics></math>. Under the assumption that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>C</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{C}</annotation></semantics></math> is well-powered, the category of subobjects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sub</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>↪</mo><msub><mi>Mod</mi> <mi>S</mi></msub><mo stretchy="false">/</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">Sub(S) \hookrightarrow Mod_S/S</annotation></semantics></math> is a (small) sup-lattice. Under the regularity assumption on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>C</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{C}</annotation></semantics></math>, the subcategory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sub</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>↪</mo><msub><mi>Mod</mi> <mi>S</mi></msub><mo stretchy="false">/</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">Sub(S) \hookrightarrow Mod_S/S</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/reflective+subcategory">reflective</a>, and by applying the reflector to the monoidal product on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>S</mi></msub><mo stretchy="false">/</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">Mod_S/S</annotation></semantics></math>, we obtain a product on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sub</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sub(S)</annotation></semantics></math> which preserves arbitrary joins in each variable, hence a quantale. The unit of the quantale is the top element, namely <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> considered as an ideal.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sieve">sieve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+ideal+of+a+monoid">principal ideal of a monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ideal+of+a+ring">ideal of a ring</a></p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on May 6, 2022 at 04:34:13. 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