Editing Syntactic monoid (section)

==Examples==
* Let <math>L</math> be the language over <math>A = \{a, b\}</math> of words of even length.  The syntactic congruence has two classes, <math>L</math> itself and <math>L_1</math>, the words of odd length.  The syntactic monoid is the group of order 2 on <math>\{L, L_1\}</math>.<ref name=S54>Straubing (1994) p.54</ref>  
* For the language <math>(ab+ba)^*</math>, the minimal automaton has 4 states and the syntactic monoid has 15 elements.<ref name=Law211>Lawson (2004) pp.211-212</ref>
* The [[bicyclic monoid]] is the syntactic monoid of the [[Dyck language]] (the language of balanced sets of parentheses).
* The [[free monoid]] on <math>A</math> (where <math>\left|A\right| > 1</math>) is the syntactic monoid of the language <math>\{ww^R \mid w \in A^*\}</math>, where <math>w^R</math> is the reversal of the word <math>w</math>.  (For <math>\left|A\right| = 1</math>, one can use the language of square powers of the letter.)
* Every non-trivial finite monoid is homomorphic{{clarify|Which way does the homorphism go? Is it onto?|date=June 2016}} 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=[https://archive.org/details/CounterFre_00_McNa/page/48 48] | url-access=registration | url=https://archive.org/details/CounterFre_00_McNa/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 regular language.<ref name=MP48/>
* The language over <math>\{a, b\}</math> in which the number of occurrences of <math>a</math> and <math>b</math> are congruent modulo <math>2^n</math> is a group language with syntactic monoid <math>\mathbb{Z} / 2^n\mathbb{Z}</math>.<ref name=Sak342/>
* [[Trace monoid]]s are examples of syntactic monoids.
* [[Marcel-Paul Schützenberger]]<ref>{{cite journal | author=Marcel-Paul Schützenberger | author-link=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 | doi=10.1016/s0019-9958(65)90108-7| doi-access=free }}</ref> characterized [[star-free language]]s as those with finite [[Aperiodic monoid|aperiodic]] syntactic monoids.<ref name=S60>Straubing (1994) p.60</ref>