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| In [[mathematics]] and [[computer science]], the '''syntactic monoid''' ''M''(''L'') of a [[formal language]] ''L'' is the smallest [[monoid]] that [[recognizable set|recognizes]] the language ''L''.
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| ==Syntactic quotient==
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| Given <math>S\subset M</math> of a monoid ''M'' of every string over some alphabet, one may define sets that consist of formal left or right [[Inverse element#In an unital magma|inverses of elements]] in ''S''. These are called [[string operations|quotients]], and one may define right or left quotients, depending on which side one is concatenating. Thus, the '''right quotient''' of ''S'' by an element <math>m\in M</math> is the set
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| :<math>S \ / \ m=\{u\in M \;\vert\; um\in S \}.</math>
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| Similarly, the '''left quotient''' is
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| :<math>m \setminus S=\{u\in M \;\vert\; mu\in S \}.</math>
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| ==Syntactic equivalence==
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| The syntactic quotient induces an [[equivalence relation]] on ''M'', called the '''syntactic relation''', or '''syntactic equivalence''' (induced by ''S''). The right syntactic equivalence is the equivalence relation
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| :<math>\sim_S \;= \{(s,t)\in M\times M \,\vert\; S \ / \ s = S \ / \ t \}.</math>
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| Similarly, the left syntactic relation is
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| :<math>\,_S\sim \;= \{(s,t)\in M\times M \,\vert\; s\setminus S = t \setminus S \}.</math>
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| The '''syntactic congruence'''{{clarify|reason='Syntactic congruence' and 'right syntactic equivalence' share the same symbolic notation. Please indicate clearly if the are meant to be identical notions (which is not obvious from their definitions). Otherwise, use different symbolic notations, and comment on the relation between both concepts, if appropriate.|date=August 2013}} may be defined as<ref name=Law210>Lawson (2004) p.210</ref>
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| :<math>u \sim_S v \Leftrightarrow \forall x, y\in M (xuy \in S \Leftrightarrow xvy \in S).</math>
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| The definition extends to a congruence defined by a subset ''S'' of a general monoid ''M''. A '''disjunctive set''' is a subset ''S'' such that the syntactic congruence defined by ''S'' is the equality relation.<ref name=Law232>Lawson (2004) p.232</ref>
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| ==Syntactic monoid==
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| The syntactic quotient is [[Quotient algebra|compatible]] with concatenation in the monoid, in that one has
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| :<math>(M \ / \ s) \ / \ t = M \ / \ (ts)</math>
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| for all <math>s,t\in M</math> (and similarly for the left quotient). Thus, the syntactic quotient is a [[monoid morphism]], and induces a [[quotient monoid]]
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| :<math>M(S)= M \ / \ \sim_S.</math> | |
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| This monoid <math>M(S)</math> is called the '''syntactic monoid''' of ''S''.
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| It can be shown that it is the smallest [[monoid]] that [[recognizable set|recognizes]] ''S'' ; that is, ''M''(''S'') recognizes ''S'', and for every monoid ''N'' recognizing ''S'', ''M''(''S'') is a quotient of a [[submonoid]] of ''N''. The syntactic monoid of ''S'' is also the [[transition monoid]] of the [[minimal automaton]] of ''S''.<ref name=Law210/><ref name=S55>Straubing (1994) p.55</ref>
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| Similarly, a language ''L'' is regular if and only if the family of quotients
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| :<math>\{m \setminus L \,\vert\; m\in M\}</math>
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| is finite. The proof showing equivalence is quite easy. Assume that a string ''x'' is read by a [[deterministic finite automaton]], with the machine proceeding into state ''p''. If ''y'' is another string read by the machine, also terminating in the same state ''p'', then clearly one has <math>x \setminus L\,= y \setminus L</math>. Thus, the number of elements in <math>\{m \setminus L \,\vert\; m\in M\}</math> is just exactly equal to the number of states of the automaton and <math>\{m \setminus L \,\vert\; m\in L\}</math> is equal to number of final states. Assume the converse: that the number of elements in <math>\{m \setminus L \,\vert\; m\in M\}</math> is finite. One can then construct an automaton where <math>Q=\{m \setminus L \,\vert\; m\in M\}</math> is the set of states, <math>F=\{m \setminus L \,\vert\; m\in L\}</math> is the set of final states, the language ''L'' is the initial state, and the transition function is given by <math>y\setminus(x \setminus L) =(xy) \setminus L</math>. Clearly, this automaton recognizes ''L''. Thus, a language ''L'' is recognizable if and only if the set <math>\{m \setminus L \,\vert\; m\in M\}</math> is finite.
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| Given a [[regular expression]] ''E'' representing ''S'', it is easy to compute the syntactic monoid of ''S''.
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| A '''group language''' is one for which the syntactic monoid is a [[Group (mathematics)|group]].<ref name=Sak342>Sakarovitch (2009) p.342</ref>
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| ==Examples==
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| * Let ''L'' be the language over ''A'' = {''a'',''b''} of words of even length. The syntactic congruence has two classes, ''L'' itself and ''L''<sub>1</sub>, the words of odd length. The syntactic monoid is the group of order 2 on {''L'',''L''<sub>1</sub>}.<ref name=S54>Straubing (1994) p.54</ref>
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| * The [[bicyclic monoid]] is the syntactic monoid of the [[Dyck language]] (the language of balanced sets of parentheses).
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| * The [[free monoid]] on ''A'' is the syntactic monoid of the language { ''ww''<sup>R</sup> | ''w'' in ''A''* }, where ''w''<sup>R</sup> denotes the reversal of word ''w''.
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| * Every finite monoid is homomorphic to the syntactic monoid of some non-trivial language,<ref name=MP48>{{cite book | last1=McNaughton | first1=Robert | last2 = Papert | first2=Seymour | author2-link=Seymour Papert | others=With an appendix by William Henneman | series=Research Monograph | volume=65 | year=1971 | title=Counter-free Automata | publisher=MIT Press | isbn=0-262-13076-9 | zbl=0232.94024 | page=48 }}</ref> but not every finite monoid is isomorphic to a syntactic monoid.<ref name=Law233>Lawson (2004) p.233</ref>
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| * Every finite group is isomorphic to the syntactic monoid of some non-trivial language.<ref name=MP48/>
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| * The language over {''a'',''b''} in which the number of occurrences of ''a'' and ''b'' are congruent modulo 2<sup>''n''</sup> is a group language with syntactic monoid '''Z'''/2<sup>''n''</sup>.<ref name=Sak342/>
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| * [[Trace monoid]]s are examples of syntactic monoids.
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| * [[Marcel-Paul Schützenberger]]<ref>{{cite journal | author=[[Marcel-Paul Schützenberger]] | title=On finite monoids having only trivial subgroups | journal=Information and Computation| year=1965| volume=8 | issue=2 | pages=190–194|url=http://igm.univ-mlv.fr/~berstel/Mps/Travaux/A/1965-4TrivialSubgroupsIC.pdf}}</ref> characterized [[star-free language]]s as those with finite [[Aperiodic_monoid|aperiodic]] syntactic monoids.<ref name=S60>Straubing (1994) p.60</ref>
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| ==References==
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| {{reflist}}
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| * {{cite book | last=Anderson | first=James A. | title=Automata theory with modern applications | others=With contributions by Tom Head | location=Cambridge | publisher=[[Cambridge University Press]] | year=2006 | isbn=0-521-61324-8 | zbl=1127.68049 }}
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| * {{cite book | last=Lawson | first=Mark V. | title=Finite automata | publisher=Chapman and Hall/CRC | year=2004 | isbn=1-58488-255-7 | zbl=1086.68074 }}
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| * {{cite book | editor1-last=Rozenberg | editor1-first=G. | editor2-last=Salomaa | editor2-first=A. | first=Jean-Eric | last=Pin | url=http://www.liafa.jussieu.fr/~jep/PDF/HandBook.pdf | chapter=10. Syntactic semigroups | title=Handbook of Formal Language Theory | volume=1 | publisher=[[Springer-Verlag]] | year=1997 | pages=679-746 | zbl=0866.68057 }}
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| * {{cite book | last=Sakarovitch | first=Jacques | title=Elements of automata theory | others=Translated from the French by Reuben Thomas | publisher=[[Cambridge University Press]] | year=2009 | isbn=978-0-521-84425-3 | zbl=1188.68177 }}
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| * {{cite book | last=Straubing | first=Howard | title=Finite automata, formal logic, and circuit complexity | series=Progress in Theoretical Computer Science | location=Basel | publisher=Birkhäuser | year=1994 | isbn=3-7643-3719-2 | zbl=0816.68086 }}
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| [[Category:Formal languages]]
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| [[Category:Semigroup theory]]
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