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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"> <p class="show_diff"> Showing changes from revision #10 to #11: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='algebra'>Algebra</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/higher+algebra'>higher algebra</a></strong></p> <p><a class='existingWikiWord' href='/nlab/show/diff/universal+algebra'>universal algebra</a></p> <h2 id='algebraic_theories'>Algebraic theories</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/algebraic+theory'>algebraic theory</a> / <a class='existingWikiWord' href='/nlab/show/diff/2-Lawvere+theory'>2-algebraic theory</a> / <a class='existingWikiWord' href='/nlab/show/diff/%28%E2%88%9E%2C1%29-algebraic+theory'>(∞,1)-algebraic theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monad'>monad</a> / <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-monad'>(∞,1)-monad</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/operad'>operad</a> / <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%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/diff/algebra+over+a+monad'>algebra over a monad</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/infinity-algebra+over+an+%28infinity%2C1%29-monad'>∞-algebra over an (∞,1)-monad</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/algebra+over+a+Lawvere+theory'>algebra over an algebraic theory</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/infinity-algebra+over+an+%28infinity%2C1%29-algebraic+theory'>∞-algebra over an (∞,1)-algebraic theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/algebra+over+an+operad'>algebra over an operad</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/infinity-algebra+over+an+%28infinity%2C1%29-operad'>∞-algebra over an (∞,1)-operad</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/action'>action</a>, <a class='existingWikiWord' href='/nlab/show/diff/infinity-action'>∞-action</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/representation'>representation</a>, <a class='existingWikiWord' href='/nlab/show/diff/infinity-representation'>∞-representation</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/module'>module</a>, <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-module'>∞-module</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/associated+bundle'>associated bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/associated+infinity-bundle'>associated ∞-bundle</a></p> </li> </ul> <h2 id='higher_algebras'>Higher algebras</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monoidal+%28infinity%2C1%29-category'>monoidal (∞,1)-category</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/symmetric+monoidal+%28infinity%2C1%29-category'>symmetric monoidal (∞,1)-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monoid+in+a+monoidal+%28infinity%2C1%29-category'>monoid in an (∞,1)-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/commutative+monoid+in+a+symmetric+monoidal+%28infinity%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/diff/smash+product+of+spectra'>smash product of spectra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/symmetric+smash+product+of+spectra'>symmetric monoidal smash product of spectra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/ring+spectrum'>ring spectrum</a>, <a class='existingWikiWord' href='/nlab/show/diff/module+spectrum'>module spectrum</a>, <a class='existingWikiWord' href='/nlab/show/diff/algebra+spectrum'>algebra spectrum</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/A-infinity-algebra'>A-∞ algebra</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/A-infinity-ring'>A-∞ ring</a>, <a class='existingWikiWord' href='/nlab/show/diff/A-infinity-space'>A-∞ space</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/C_%E2%88%9E-algebra'>C-∞ algebra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/E-infinity-ring'>E-∞ ring</a>, <a class='existingWikiWord' href='/nlab/show/diff/E-infinity+algebra'>E-∞ algebra</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-module'>∞-module</a>, <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-module+bundle'>(∞,1)-module bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/multiplicative+cohomology+theory'>multiplicative cohomology theory</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/L-infinity-algebra'>L-∞ algebra</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/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/diff/model+structure+on+simplicial+algebras'>model structure on simplicial T-algebras</a> / <a class='existingWikiWord' href='/nlab/show/diff/homotopy+T-algebra'>homotopy T-algebra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+operads'>model structure on operads</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/Isbell+duality'>Isbell duality</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/derived+geometry'>derived geometry</a></p> </li> </ul> <h2 id='theorems'>Theorems</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Deligne+conjecture'>Deligne conjecture</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/delooping+hypothesis'>delooping hypothesis</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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> <h4 id='monoid_theory'>Monoid theory</h4> <div class='hide'> <p><strong>monoid theory</strong> in <a class='existingWikiWord' href='/nlab/show/diff/algebra'>algebra</a>:</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monoid'>monoid</a>, <a class='existingWikiWord' href='/nlab/show/diff/monoid+in+a+monoidal+%28infinity%2C1%29-category'>infinity-monoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monoid+in+a+monoidal+category'>monoid object</a>, <a class='existingWikiWord' href='/nlab/show/diff/monoid+in+a+monoidal+%28infinity%2C1%29-category'>monoid object in an (infinity,1)-category</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/semiring'>semiring</a>, <a class='existingWikiWord' href='/nlab/show/diff/rig'>rig</a>, <a class='existingWikiWord' href='/nlab/show/diff/ring'>ring</a>, <a class='existingWikiWord' href='/nlab/show/diff/associative+unital+algebra'>associative unital algebra</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/category+of+monoids'>Mon</a>, <a class='existingWikiWord' href='/nlab/show/diff/category+of+monoids'>CMon</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homomorphism'>monoid homomorphism</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/trivial+monoid'>trivial monoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/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/diff/inverse'>inverse element</a>, <a class='existingWikiWord' href='/nlab/show/diff/unit'>unit</a>, <a class='existingWikiWord' href='/nlab/show/diff/irreducible+element'>irreducible element</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/ideal+in+a+monoid'>ideal in a monoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/principal+ideal+of+a+monoid'>principal ideal in a monoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/commutative+monoid'>commutative monoid</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/tensor+product+of+commutative+monoids'>tensor product of commutative monoids</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cancellative+monoid'>cancellative monoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/GCD+monoid'>GCD monoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/unique+factorization+monoid'>unique factorization monoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/B%C3%A9zout+monoid'>Bézout monoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/principal+ideal+monoid'>principal ideal monoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/group'>group</a>, <a class='existingWikiWord' href='/nlab/show/diff/abelian+group'>abelian group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/absorption+monoid'>absorption monoid</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/zero-divisor'>zero divisor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/integral+monoid'>integral monoid</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/free+monoid'>free monoid</a>, <a class='existingWikiWord' href='/nlab/show/diff/free+commutative+monoid'>free commutative monoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/graphic+category'>graphic monoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/action'>monoid action</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/module+over+a+monoid'>module over a monoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/localization+of+a+monoid'>localization of a monoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/group+completion'>group completion</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/endomorphism'>endomorphism monoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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> <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><ul><li><a href='#ideals_forming_a_quantale'>Ideals forming a quantale</a></li></ul></li><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/diff/monoid'>monoid</a> (or <a class='existingWikiWord' href='/nlab/show/diff/semigroup'>semigroup</a>) <math class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math>, a <strong>left ideal</strong> in <math class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> is <a class='existingWikiWord' href='/nlab/show/diff/subset'>subset</a> <math class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi><mi>A</mi></mrow><annotation encoding='application/x-tex'>S A</annotation></semantics></math> is contained in <math class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_6' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_8' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> is a subset <math class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_10' xmlns='http://www.w3.org/1998/Math/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/diff/monoidal+category'>monoidal category</a> <math class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_11' xmlns='http://www.w3.org/1998/Math/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/diff/monoid+in+a+monoidal+category'>monoid object</a> (or semigroup object) <math class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>, we can <a class='existingWikiWord' href='/nlab/show/diff/internalization'>internalise</a> the above. For instance, if <math class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_14' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_15' xmlns='http://www.w3.org/1998/Math/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/diff/subobject'>subobject</a> <math class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math>, i.e., a mono <math class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_18' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_19' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='block' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_20' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_21' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_22' xmlns='http://www.w3.org/1998/Math/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/diff/submonoid'>submonoid</a>, inasmuch as the monoid unit <math class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_23' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_24' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo>=</mo></mrow><annotation encoding='application/x-tex'>C = </annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/Ab'>Ab</a>, a monoid in <math class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/ring'>ring</a> and the corresponding notion of <em><a class='existingWikiWord' href='/nlab/show/diff/ideal'>ideal in a ring</a></em> is the most common notion of ideal.</p> <p>See <a class='existingWikiWord' href='/nlab/show/diff/ideal'>ideal</a> for ideals in more well known contexts: commutative idempotent monoids (<a class='existingWikiWord' href='/nlab/show/diff/semilattice'>semilattices</a>) and monoids in <a class='existingWikiWord' href='/nlab/show/diff/Ab'>Ab</a> (<a class='existingWikiWord' href='/nlab/show/diff/ring'>rings</a>).</p> <h2 id='properties_and_constructions'>Properties and constructions</h2> <p>An ideal <math class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_27' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_28' xmlns='http://www.w3.org/1998/Math/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/diff/submonoid'>submonoid</a> iff <math class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_29' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_30' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_31' xmlns='http://www.w3.org/1998/Math/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/diff/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/diff/ring'>ring</a> <math class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>, the product <math class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi><mi>J</mi></mrow><annotation encoding='application/x-tex'>I J</annotation></semantics></math> of ideals <math class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_34' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_35' xmlns='http://www.w3.org/1998/Math/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/diff/residuated+lattice'>residuated lattice</a>, in that there are also division operations where</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_36' xmlns='http://www.w3.org/1998/Math/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' /><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/diff/adjunction'>adjointness</a> relations: <math class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_37' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_38' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_39' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_40' xmlns='http://www.w3.org/1998/Math/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/diff/well-powered+category'>well-powered</a> <a class='existingWikiWord' href='/nlab/show/diff/regular+category'>regular</a> <a class='existingWikiWord' href='/nlab/show/diff/cosmos'>cosmos</a> (‘cosmos’ in the sense of complete cocomplete symmetric monoidal closed category). Just using the fact that <math class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_41' xmlns='http://www.w3.org/1998/Math/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/diff/monoidal+bicategory'>monoidal bicategory</a> <math class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_42' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_44' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_45' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> right-<math class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi></mrow><annotation encoding='application/x-tex'>T</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/module'>modules</a>, and whose 2-morphisms are bimodule homomorphisms.</p> <p>For each monoid <math class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math>, there is a subbicategory of <math class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_49' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_50' xmlns='http://www.w3.org/1998/Math/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/diff/closed+monoidal+category'>biclosed monoidal category</a> <math class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_51' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_52' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_53' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> with its standard <math class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_55' xmlns='http://www.w3.org/1998/Math/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/diff/over+category'>slice</a> <math class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_56' xmlns='http://www.w3.org/1998/Math/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/diff/semicartesian+monoidal+category'>semicartesian monoidal category</a>) forms another complete and cocomplete biclosed monoidal category.</p> <p>An ideal of <math class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> is just a subobject of <math class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_58' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_59' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_60' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_61' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_62' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>C</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{C}</annotation></semantics></math>, the subcategory <math class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_63' xmlns='http://www.w3.org/1998/Math/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/diff/reflective+subcategory'>reflective</a>, and by applying the reflector to the monoidal product on <math class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_64' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_65' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_8ee5f6c4069df508666509e2e79b5fcf659ef663_66' xmlns='http://www.w3.org/1998/Math/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/diff/sieve'>sieve</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/principal+ideal+of+a+monoid'>principal ideal of a monoid</a></p> </li><ins class='diffins'> </ins><ins class='diffins'><li> <p><a class='existingWikiWord' href='/nlab/show/diff/ideal'>ideal of a ring</a></p> </li></ins> </ul> <p> </p> <p> </p> </div> <div class="revisedby"> <p> Last revised on May 6, 2022 at 04:34:13. 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