syntactic monoid

An alphabet is a finite set.

The free monoid on a given alphabet is the monoid whose elements are all the strings of zero or more elements from that set, with string concatenation as the monoid operation and the empty string as the identity element.

Given a subset $S$ of a free monoid $M$, one may define sets that consist of formal left or right inverses of elements in $S$. These are called 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 $m$ from $M$ is the set

:$S\; \backslash \; /\; \backslash \; m=\backslash \{u\backslash in\; M\; \backslash ;\backslash vert\backslash ;\; um\backslash in\; S\; \backslash \}.$

Similarly, the left quotient is

:$m\; \backslash setminus\; S=\backslash \{u\backslash in\; M\; \backslash ;\backslash vert\backslash ;\; mu\backslash in\; S\; \backslash \}.$

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

:$s\; \backslash sim\_S\; t\; \backslash \; \backslash Leftrightarrow\backslash \; S\; \backslash ,/\; \backslash ,s\; \backslash ;=\backslash ;\; S\; \backslash ,/\; \backslash ,t\; \backslash \; \backslash Leftrightarrow\backslash \; (\backslash forall\; x\backslash in\; M\backslash colon\backslash \; xs\; \backslash in\; S\; \backslash Leftrightarrow\; xt\; \backslash in\; S)$.

Similarly, the left syntactic equivalence is

:$s\; \backslash ;\{\}\_S\{\backslash sim\}\backslash ;\; t\; \backslash \; \backslash Leftrightarrow\backslash \; s\; \backslash setminus\; S\; \backslash ;=\backslash ;\; t\; \backslash setminus\; S\; \backslash \; \backslash Leftrightarrow\backslash \; (\backslash forall\; y\backslash in\; M\backslash colon\backslash \; sy\; \backslash in\; S\; \backslash Leftrightarrow\; ty\; \backslash in\; S)$.

Observe that the right syntactic equivalence is a left congruence with respect to string concatenation and vice versa; i.e., $s\; \backslash sim\_S\; t\; \backslash \; \backslash Rightarrow\backslash \; xs\; \backslash sim\_S\; xt\backslash$ for all $x\; \backslash in\; M$.

The syntactic congruence or Myhill congruenceHolcombe (1982) p.160 is defined asLawson (2004) p.210

:$s\; \backslash equiv\_S\; t\; \backslash \; \backslash Leftrightarrow\backslash \; (\backslash forall\; x,\; y\backslash in\; M\backslash colon\backslash \; xsy\; \backslash in\; S\; \backslash Leftrightarrow\; xty\; \backslash in\; S)$.

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.Lawson (2004) p.232

Let us call $[s]\_S$ the equivalence class of $s$ for the syntactic congruence.

The syntactic congruence is compatible with concatenation in the monoid, in that one has

:$[s]\_S[t]\_S=[st]\_S$

for all $s,t\backslash in\; M$. Thus, the syntactic quotient is a monoid morphism, and induces a quotient monoid

:$M(S)=\; M\; \backslash \; /\; \backslash \; \{\backslash equiv\_S\}$.

This monoid $M(S)$ is called the syntactic monoid of $S$.

It can be shown that it is the smallest monoid that 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 $S$.Straubing (1994) p.55

A group language is one for which the syntactic monoid is a group.Sakarovitch (2009) p.342

• Let $L$ be the language over $A\; =\; \backslash \{a,\; b\backslash \}$ of words of even length. The syntactic congruence has two classes, $L$ itself and $L\_1$, the words of odd length. The syntactic monoid is the group of order 2 on $\backslash \{L,\; L\_1\backslash \}$.Straubing (1994) p.54
• For the language $(ab+ba)^*$, the minimal automaton has 4 states and the syntactic monoid has 15 elements.Lawson (2004) pp.211-212
• The bicyclic monoid is the syntactic monoid of the Dyck language (the language of balanced sets of parentheses).
• The free monoid on $A$ (where $\backslash left|A\backslash right|\; >\; 1$) is the syntactic monoid of the language $\backslash \{ww^R\; \backslash mid\; w\; \backslash in\; A^*\backslash \}$, where $w^R$ is the reversal of the word $w$. (For $\backslash left|A\backslash right|\; =\; 1$, 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,{{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 }} but not every finite monoid is isomorphic to a syntactic monoid.Lawson (2004) p.233
• Every finite group is isomorphic to the syntactic monoid of some regular language.
• The language over $\backslash \{a,\; b\backslash \}$ in which the number of occurrences of $a$ and $b$ are congruent modulo $2^n$ is a group language with syntactic monoid $\backslash mathbb\{Z\}\; /\; 2^n\backslash mathbb\{Z\}$.
• Trace monoids are examples of syntactic monoids.
• Marcel-Paul Schützenberger{{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 }} characterized star-free languages as those with finite aperiodic syntactic monoids.Straubing (1994) p.60

{{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=Holcombe | first=W.M.L. | title=Algebraic automata theory | zbl=0489.68046 | series=Cambridge Studies in Advanced Mathematics | volume=1 | publisher=Cambridge University Press | year=1982 | isbn=0-521-60492-3 }}
• {{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-Éric | last=Pin |author-link = Jean-Éric 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 | url=https://archive.org/details/finiteautomatafo0000stra | url-access=registration | 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