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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''.~~		Eusebio is the name people use to call me and my friends and I think getting this done sounds quite good when you say it. Idaho is our birth city. I second-hand to be unemployed but now I am a major cashier. My partners say it's not [http://en.search.wordpress.com/?q=incredibly incredibly] good for me but the I love doing may be to bake but I'm so [http://www.Encyclopedia.com/searchresults.aspx?q=thinking thinking] on starting new things. I'm not good at web design but you might desire to check my website: http://prometeu.net<br><br>my website :: clash of clans hack cydia ([http://prometeu.net click the up coming article])

	~~==Syntactic quotient==~~
	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~~
	~~:<math>S \ / \ m=\{u\in M \;\vert\; um\in S \}.</math>~~

	~~Similarly, the '''left quotient''' is~~

	~~:<math>m \setminus S=\{u\in M \;\vert\; mu\in S \}.</math>~~

	~~==Syntactic equivalence==~~
	~~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~~

	~~:<math>\sim_S \;= \{(s,t)\in M\times M \,\vert\; S \ / \ s = S \ / \ t \}~~.~~</math>~~

	~~Similarly, the left syntactic relation~~ is

	~~:<math>\,_S\sim \;= \{(s,t)\in M\times M \,\vert\; s\setminus S = t \setminus S \}~~.~~</math>~~

	~~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>

	~~:<math>u \sim_S v \Leftrightarrow \forall x, y\in M (xuy \in S \Leftrightarrow xvy \in S).</math>~~

	~~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>~~

	~~==Syntactic monoid==~~
	~~The syntactic quotient is [[Quotient algebra\|compatible]] with concatenation in the monoid, in that one has~~

	~~:<math>(M \ / \~~ s~~) \ / \ t = M \ / \ (ts)</math>~~

	~~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]]~~

	:~~<math>M(S)= M \~~ / ~~\ \sim_S.</math>~~

	~~This monoid <math>M(S)<~~/~~math> is called the '''syntactic monoid''' of ''S''~~.
	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>~~

	~~Similarly, a language ''L'' is regular if and only if the family of quotients~~

	~~:<math>\{m \setminus L \,\vert\; m\in M\}</math>~~

	~~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~~.

	~~Given a [[regular expression]] ''E'' representing ''S'', it is easy to compute the syntactic monoid of ''S''.~~

	~~A '''group language''' is one for which the syntactic monoid is a [[Group (mathematics)\|group]~~]~~.<ref name=Sak342>Sakarovitch (2009) p.342</ref>~~

	~~==Examples==~~
	* 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>~~
	* The [[bicyclic monoid]] is the syntactic monoid of the [[Dyck language]] (the language of balanced sets of parentheses).
	* 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''.~~
	* 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>~~
	* Every finite group is isomorphic to the syntactic monoid of some non-trivial language.<ref name=MP48/>
	* 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/>
	* [[Trace monoid]]s are examples of syntactic monoids.
	* [[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>~~

	~~==References==~~
	~~{{reflist}}~~
	* {{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 }}~~
	* {{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 }}
	* {{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 }}~~
	* {{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 }}~~
	* {{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 }}

	~~[[Category:Formal languages]]~~
	~~[[Category:Semigroup theory]~~]

2.51.159.244 at 17:03, 18 January 2014

2014-01-18T17:03:18Z

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en>Crowsnest: Undid revision 509912627 by 188.26.22.131 (talk) he did his PhD thesis on capillary waves, but it is only one of many contributions to the theory of capillary waves

2012-08-30T11:01:12Z

Undid revision 509912627 by 188.26.22.131 (talk) he did his PhD thesis on capillary waves, but it is only one of many contributions to the theory of capillary waves

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Capillary wave - Revision history

en>YiFeiBot: Bot: Migrating interwiki links, now provided by Wikidata on d:q15143885

2.51.159.244 at 17:03, 18 January 2014

en>Crowsnest: Undid revision 509912627 by 188.26.22.131 (talk) he did his PhD thesis on capillary waves, but it is only one of many contributions to the theory of capillary waves