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Syntactic monoid

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In <a href="Mathematics" title="Mathematics">mathematics</a> and <a href="Computer_science" title="Computer science">computer science</a>, the syntactic monoid <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M(L)}">

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<annotation encoding="application/x-tex">{\displaystyle M(L)}</annotation>

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</math><img src="./76f259407b5e0bacf2947f1ed335db19f68487f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.834ex; height:2.843ex;" alt="{\displaystyle M(L)}" loading="lazy"> of a <a href="Formal_language" title="Formal language">formal language</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

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<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

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</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> is the minimal <a href="Monoid" title="Monoid">monoid</a> that <a href="Recognizable_set" title="Recognizable set">recognizes</a> the language <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

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<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

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</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy">. By the <a href="Myhill%E2%80%93Nerode_theorem" title="Myhill–Nerode theorem">Myhill–Nerode theorem</a>, the syntactic monoid is unique up to unique isomorphism.

<div class="mw-heading mw-heading2"><h2 id="Syntactic_quotient">Syntactic quotient</h2></div>

An <a href="Alphabet_(formal_languages)" title="Alphabet (formal languages)">alphabet</a> is a finite <a href="Set_(mathematics)" title="Set (mathematics)">set</a>.

The <a href="Free_monoid" title="Free monoid">free monoid</a> on a given alphabet is the monoid whose elements are all the <a href="String_(computer_science)" title="String (computer science)">strings</a> of zero or more elements from that set, with <a href="String_concatenation" class="mw-redirect" title="String concatenation">string concatenation</a> as the monoid operation and the <a href="Empty_string" title="Empty string">empty string</a> as the <a href="Identity_element" title="Identity element">identity element</a>.

Given a <a href="Subset" title="Subset">subset</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

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</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> of a free monoid <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">, one may define sets that consist of formal left or right <a href="Inverse_element#In_a_unital_magma" title="Inverse element">inverses of elements</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

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</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy">. These are called <a href="Quotient_of_a_formal_language" title="Quotient of a formal language">quotients</a>, and one may define right or left quotients, depending on which side one is concatenating. Thus, the right quotient of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

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</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> by an element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

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<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

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</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> is the set

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<annotation encoding="application/x-tex">{\displaystyle S\ /\ m=\{u\in M\;\vert \;um\in S\}.}</annotation>

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</math><img src="./931bf0602a50e8aea22fc7dbd8206b907894f676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.193ex; height:2.843ex;" alt="{\displaystyle S\ /\ m=\{u\in M\;\vert \;um\in S\}.}" loading="lazy"></dd></dl>

Similarly, the left quotient is

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<annotation encoding="application/x-tex">{\displaystyle m\setminus S=\{u\in M\;\vert \;mu\in S\}.}</annotation>

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</math><img src="./bc15a96700328875aa61b98ae6abdbc6caf49310.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.064ex; height:2.843ex;" alt="{\displaystyle m\setminus S=\{u\in M\;\vert \;mu\in S\}.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Syntactic_equivalence">Syntactic equivalence</h2></div>

The syntactic quotient induces an <a href="Equivalence_relation" title="Equivalence relation">equivalence relation</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">, called the syntactic relation, or syntactic equivalence (induced by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

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</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy">).

The right syntactic equivalence is the equivalence relation

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<annotation encoding="application/x-tex">{\displaystyle s\sim _{S}t\ \Leftrightarrow \ S\,/\,s\;=\;S\,/\,t\ \Leftrightarrow \ (\forall x\in M\colon \ xs\in S\Leftrightarrow xt\in S)}</annotation>

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</math><img src="./069287a87a0f54f49ad57c91a6c4085945762e66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:57.275ex; height:2.843ex;" alt="{\displaystyle s\sim _{S}t\ \Leftrightarrow \ S\,/\,s\;=\;S\,/\,t\ \Leftrightarrow \ (\forall x\in M\colon \ xs\in S\Leftrightarrow xt\in S)}" loading="lazy">.</dd></dl>

Similarly, the left syntactic equivalence is

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<annotation encoding="application/x-tex">{\displaystyle s\;{}_{S}{\sim }\;t\ \Leftrightarrow \ s\setminus S\;=\;t\setminus S\ \Leftrightarrow \ (\forall y\in M\colon \ sy\in S\Leftrightarrow ty\in S)}</annotation>

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</math><img src="./d87e7f235fed0429fab3141e3a2c6678b8d1acb0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:57.268ex; height:2.843ex;" alt="{\displaystyle s\;{}_{S}{\sim }\;t\ \Leftrightarrow \ s\setminus S\;=\;t\setminus S\ \Leftrightarrow \ (\forall y\in M\colon \ sy\in S\Leftrightarrow ty\in S)}" loading="lazy">.</dd></dl>

Observe that the right syntactic equivalence is a left <a href="Congruence_relation" title="Congruence relation">congruence</a> with respect to <a href="String_concatenation" class="mw-redirect" title="String concatenation">string concatenation</a> and vice versa; i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\sim _{S}t\ \Rightarrow \ xs\sim _{S}xt\ }">

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<annotation encoding="application/x-tex">{\displaystyle s\sim _{S}t\ \Rightarrow \ xs\sim _{S}xt\ }</annotation>

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</math><img src="./d5e2f78f261cb3863ff2a3b974c41677a4174e04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.657ex; height:2.343ex;" alt="{\displaystyle s\sim _{S}t\ \Rightarrow \ xs\sim _{S}xt\ }" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in M}">

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<annotation encoding="application/x-tex">{\displaystyle x\in M}</annotation>

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</math><img src="./9df57d73e9532bb93a1439890bcddbc2806f5859.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.613ex; height:2.176ex;" alt="{\displaystyle x\in M}" loading="lazy">.

The syntactic congruence or <a href="John_Myhill" title="John Myhill">Myhill</a> congruence<a href="#cite_note-Hol160-1">[1]</a> is defined as<a href="#cite_note-Law210-2">[2]</a>

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<annotation encoding="application/x-tex">{\displaystyle s\equiv _{S}t\ \Leftrightarrow \ (\forall x,y\in M\colon \ xsy\in S\Leftrightarrow xty\in S)}</annotation>

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</math><img src="./f754e360517bf52b1a1212d7b87de1d542d4a06a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.809ex; height:2.843ex;" alt="{\displaystyle s\equiv _{S}t\ \Leftrightarrow \ (\forall x,y\in M\colon \ xsy\in S\Leftrightarrow xty\in S)}" loading="lazy">.</dd></dl>

The definition extends to a congruence defined by a subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

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</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> of a general monoid <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">. A disjunctive set is a subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

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</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> such that the syntactic congruence defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

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</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> is the equality relation.<a href="#cite_note-Law232-3">[3]</a>

Let us call <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [s]_{S}}">

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<annotation encoding="application/x-tex">{\displaystyle [s]_{S}}</annotation>

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</math><img src="./9840432df082a10384e711f63df10ae22d25bdf9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.677ex; height:2.843ex;" alt="{\displaystyle [s]_{S}}" loading="lazy"> the equivalence class of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

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<annotation encoding="application/x-tex">{\displaystyle s}</annotation>

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</math><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"> for the syntactic congruence.

The syntactic congruence is <a href="Quotient_(universal_algebra)" title="Quotient (universal algebra)">compatible</a> with concatenation in the monoid, in that one has

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<annotation encoding="application/x-tex">{\displaystyle [s]_{S}[t]_{S}=[st]_{S}}</annotation>

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</math><img src="./2de1a348e052caf217cb4a04abf6a9aa95d07aa3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.717ex; height:2.843ex;" alt="{\displaystyle [s]_{S}[t]_{S}=[st]_{S}}" loading="lazy"></dd></dl>

for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s,t\in M}">

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<annotation encoding="application/x-tex">{\displaystyle s,t\in M}</annotation>

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</math><img src="./ae5941e435f6e9dcbb126638a83ae7fbb6d3949d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.247ex; height:2.509ex;" alt="{\displaystyle s,t\in M}" loading="lazy">. Thus, the syntactic quotient is a <a href="Monoid_morphism" class="mw-redirect" title="Monoid morphism">monoid morphism</a>, and induces a <a href="Quotient_monoid" class="mw-redirect" title="Quotient monoid">quotient monoid</a>

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<annotation encoding="application/x-tex">{\displaystyle M(S)=M\ /\ {\equiv _{S}}}</annotation>

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</math><img src="./6e107f3ee9d98efdd5b5b2d29aea0bc4dbc936c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.716ex; height:2.843ex;" alt="{\displaystyle M(S)=M\ /\ {\equiv _{S}}}" loading="lazy">.</dd></dl>

This monoid <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M(S)}">

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<annotation encoding="application/x-tex">{\displaystyle M(S)}</annotation>

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</math><img src="./5303ad5c862a6437f236231dd4293efd99a82ee5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.751ex; height:2.843ex;" alt="{\displaystyle M(S)}" loading="lazy"> is called the syntactic monoid of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

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</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy">.

It can be shown that it is the smallest <a href="Monoid" title="Monoid">monoid</a> that <a href="Recognizable_set" title="Recognizable set">recognizes</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

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</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy">; that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M(S)}">

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<annotation encoding="application/x-tex">{\displaystyle M(S)}</annotation>

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</math><img src="./5303ad5c862a6437f236231dd4293efd99a82ee5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.751ex; height:2.843ex;" alt="{\displaystyle M(S)}" loading="lazy"> recognizes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

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</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy">, and for every monoid <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

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<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

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</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> recognizing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

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</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M(S)}">

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<annotation encoding="application/x-tex">{\displaystyle M(S)}</annotation>

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</math><img src="./5303ad5c862a6437f236231dd4293efd99a82ee5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.751ex; height:2.843ex;" alt="{\displaystyle M(S)}" loading="lazy"> is a quotient of a <a href="Submonoid" class="mw-redirect" title="Submonoid">submonoid</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

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<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

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</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy">. The syntactic monoid of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

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</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> is also the <a href="Transition_monoid" class="mw-redirect" title="Transition monoid">transition monoid</a> of the <a href="Minimal_automaton" class="mw-redirect" title="Minimal automaton">minimal automaton</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

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</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy">.<a href="#cite_note-Hol160-1">[1]</a><a href="#cite_note-Law210-2">[2]</a><a href="#cite_note-S55-4">[4]</a>

A group language is one for which the syntactic monoid is a <a href="Group_(mathematics)" title="Group (mathematics)">group</a>.<a href="#cite_note-Sak342-5">[5]</a>

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

</semantics>

</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> be the language over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=\{a,b\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A=\{a,b\}}</annotation>

</semantics>

</math><img src="./469abae5074de5867f770aec58e32e30cad048d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.428ex; height:2.843ex;" alt="{\displaystyle A=\{a,b\}}" loading="lazy"> of words of even length. The syntactic congruence has two classes, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

</semantics>

</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> itself and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{1}}</annotation>

</semantics>

</math><img src="./0e79dc1b001f8b923df475ed14de023cbc456013.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.637ex; height:2.509ex;" alt="{\displaystyle L_{1}}" loading="lazy">, the words of odd length. The syntactic monoid is the group of order 2 on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{L,L_{1}\}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{L,L_{1}\}}</annotation>

</semantics>

</math><img src="./40466a953dde5094c1ac718179fb9f5321358e27.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.579ex; height:2.843ex;" alt="{\displaystyle \{L,L_{1}\}}" loading="lazy">.<a href="#cite_note-S54-6">[6]</a></li>

<li>For the language <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (ab+ba)^{*}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (ab+ba)^{*}}</annotation>

</semantics>

</math><img src="./bdf0c3a9db0514489f72e38558b1b95becbc17f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.159ex; height:2.843ex;" alt="{\displaystyle (ab+ba)^{*}}" loading="lazy">, the minimal automaton has 4 states and the syntactic monoid has 15 elements.<a href="#cite_note-Law211-7">[7]</a></li>

<li>The <a href="Bicyclic_monoid" class="mw-redirect" title="Bicyclic monoid">bicyclic monoid</a> is the syntactic monoid of the <a href="Dyck_language" title="Dyck language">Dyck language</a> (the language of balanced sets of parentheses).</li>

<li>The <a href="Free_monoid" title="Free monoid">free monoid</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> (where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|A\right|>1}">

<mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left|A\right|>1}</annotation>

</semantics>

</math><img src="./e61e3769c1aedcb595d96c7bf4047dbe6c8d6ff7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.298ex; height:2.843ex;" alt="{\displaystyle \left|A\right|>1}" loading="lazy">) is the syntactic monoid of the language <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{ww^{R}\mid w\in A^{*}\}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{ww^{R}\mid w\in A^{*}\}}</annotation>

</semantics>

</math><img src="./31a0387055ffc49a2d1477b4cedd3deb81622729.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.372ex; height:3.176ex;" alt="{\displaystyle \{ww^{R}\mid w\in A^{*}\}}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w^{R}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w^{R}}</annotation>

</semantics>

</math><img src="./2bfc73eb15807bc9a8b0f6dd120faf597cd273a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.144ex; height:2.676ex;" alt="{\displaystyle w^{R}}" loading="lazy"> is the reversal of the word <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w}</annotation>

</semantics>

</math><img src="./88b1e0c8e1be5ebe69d18a8010676fa42d7961e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}" loading="lazy">. (For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|A\right|=1}">

<mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left|A\right|=1}</annotation>

</semantics>

</math><img src="./fcc63238d29125f3d1b8ec40c1867b38c50e02ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.298ex; height:2.843ex;" alt="{\displaystyle \left|A\right|=1}" loading="lazy">, one can use the language of square powers of the letter.)</li>

<li>Every non-trivial finite monoid is homomorphic to the syntactic monoid of some non-trivial language,<a href="#cite_note-MP48-8">[8]</a> but not every finite monoid is isomorphic to a syntactic monoid.<a href="#cite_note-Law233-9">[9]</a></li>

<li>Every <a href="Finite_group" title="Finite group">finite group</a> is isomorphic to the syntactic monoid of some regular language.<a href="#cite_note-MP48-8">[8]</a></li>

<li>The language over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{a,b\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{a,b\}}</annotation>

</semantics>

</math><img src="./8127b44bf0e5a64fdc9301e188852ab9b97a1fe8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.586ex; height:2.843ex;" alt="{\displaystyle \{a,b\}}" loading="lazy"> in which the number of occurrences of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b}</annotation>

</semantics>

</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> are congruent modulo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{n}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2^{n}}</annotation>

</semantics>

</math><img src="./8226f30650ee4fe4e640c6d2798127e80e9c160d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.381ex; height:2.343ex;" alt="{\displaystyle 2^{n}}" loading="lazy"> is a group language with syntactic monoid <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /2^{n}\mathbb {Z} }">

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /2^{n}\mathbb {Z} }</annotation>

</semantics>

</math><img src="./28526fdf9d9d5ac32c95d34e9414ef6db8642b31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.644ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /2^{n}\mathbb {Z} }" loading="lazy">.<a href="#cite_note-Sak342-5">[5]</a></li>

<li><a href="Trace_monoid" title="Trace monoid">Trace monoids</a> are examples of syntactic monoids.</li>

<li><a href="Marcel-Paul_Sch%C3%BCtzenberger" title="Marcel-Paul Schützenberger">Marcel-Paul Schützenberger</a><a href="#cite_note-10">[10]</a> characterized <a href="Star-free_language" title="Star-free language">star-free languages</a> as those with finite <a href="Aperiodic_monoid" class="mw-redirect" title="Aperiodic monoid">aperiodic</a> syntactic monoids.<a href="#cite_note-S60-11">[11]</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-Hol160-1">^ <a href="#cite_ref-Hol160_1-0">a</a> <a href="#cite_ref-Hol160_1-1">b</a> Holcombe (1982) p. 160

</li>

<li id="cite_note-Law210-2">^ <a href="#cite_ref-Law210_2-0">a</a> <a href="#cite_ref-Law210_2-1">b</a> Lawson (2004) p.210

</li>

<li id="cite_note-Law232-3"><a href="#cite_ref-Law232_3-0">^</a> Lawson (2004) p.232

</li>

<li id="cite_note-S55-4"><a href="#cite_ref-S55_4-0">^</a> Straubing (1994) p.55

</li>

<li id="cite_note-Sak342-5">^ <a href="#cite_ref-Sak342_5-0">a</a> <a href="#cite_ref-Sak342_5-1">b</a> Sakarovitch (2009) p.342

</li>

<li id="cite_note-S54-6"><a href="#cite_ref-S54_6-0">^</a> Straubing (1994) p.54

</li>

<li id="cite_note-Law211-7"><a href="#cite_ref-Law211_7-0">^</a> Lawson (2004) pp.211-212

</li>

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</style><cite id="CITEREFMcNaughtonPapert1971" class="citation book cs1">McNaughton, Robert; <a href="Seymour_Papert" title="Seymour Papert">Papert, Seymour</a> (1971). <a rel="nofollow" class="external text" href="https://archive.org/details/CounterFre_00_McNa/page/48">Counter-free Automata</a>. Research Monograph. Vol. 65. With an appendix by William Henneman. MIT Press. p. <a rel="nofollow" class="external text" href="https://archive.org/details/CounterFre_00_McNa/page/48">48</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-262-13076-9</bdi>. <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0232.94024">0232.94024</a>.</cite>

</li>

<li id="cite_note-Law233-9"><a href="#cite_ref-Law233_9-0">^</a> Lawson (2004) p.233

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> <cite id="CITEREFMarcel-Paul_Schützenberger1965" class="citation journal cs1"><a href="Marcel-Paul_Sch%C3%BCtzenberger" title="Marcel-Paul Schützenberger">Marcel-Paul Schützenberger</a> (1965). <a rel="nofollow" class="external text" href="http://igm.univ-mlv.fr/~berstel/Mps/Travaux/A/1965-4TrivialSubgroupsIC.pdf">"On finite monoids having only trivial subgroups"</a> (PDF). Information and Computation. 8 (2): 190–194. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fs0019-9958%2865%2990108-7">10.1016/s0019-9958(65)90108-7</a>.</cite>

</li>

<li id="cite_note-S60-11"><a href="#cite_ref-S60_11-0">^</a> Straubing (1994) p.60

</li>

</ol></div></div>

<ul><li><cite id="CITEREFAnderson2006" class="citation book cs1">Anderson, James A. (2006). Automata theory with modern applications. With contributions by Tom Head. Cambridge: <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-61324-8</bdi>. <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1127.68049">1127.68049</a>.</cite></li>

<li><cite id="CITEREFHolcombe1982" class="citation book cs1">Holcombe, W.M.L. (1982). Algebraic automata theory. Cambridge Studies in Advanced Mathematics. Vol. 1. <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-60492-3</bdi>. <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0489.68046">0489.68046</a>.</cite></li>

<li><cite id="CITEREFLawson2004" class="citation book cs1">Lawson, Mark V. (2004). Finite automata. Chapman and Hall/CRC. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>1-58488-255-7</bdi>. <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1086.68074">1086.68074</a>.</cite></li>

<li><cite id="CITEREFPin1997" class="citation book cs1"><a href="Jean-%C3%89ric_Pin" title="Jean-Éric Pin">Pin, Jean-Éric</a> (1997). "10. Syntactic semigroups". In Rozenberg, G.; Salomaa, A. (eds.). <a rel="nofollow" class="external text" href="http://www.liafa.jussieu.fr/~jep/PDF/HandBook.pdf">Handbook of Formal Language Theory</a> (PDF). Vol. 1. <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. pp. 679–746. <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0866.68057">0866.68057</a>.</cite></li>

<li><cite id="CITEREFSakarovitch2009" class="citation book cs1">Sakarovitch, Jacques (2009). Elements of automata theory. Translated from the French by Reuben Thomas. <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-84425-3</bdi>. <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1188.68177">1188.68177</a>.</cite></li>

<li><cite id="CITEREFStraubing1994" class="citation book cs1">Straubing, Howard (1994). <a rel="nofollow" class="external text" href="https://archive.org/details/finiteautomatafo0000stra">Finite automata, formal logic, and circuit complexity</a>. Progress in Theoretical Computer Science. Basel: Birkhäuser. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-7643-3719-2</bdi>. <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0816.68086">0816.68086</a>.</cite></li></ul></div><div><div class="zim-footer">

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