Syntactic monoid


In mathematics and computer science, the syntactic monoid M of a formal language L is the smallest monoid that recognizes the language L.

Syntactic quotient

The free monoid on a given set 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 of a free monoid, one may define sets that consist of formal left or right inverses of elements in. These are called quotients, and one may define right or left quotients, depending on which side one is concatenating. Thus, the right quotient of by an element from is the set
Similarly, the left quotient is

Syntactic equivalence

The syntactic quotient induces an equivalence relation on M, called the syntactic relation, or syntactic equivalence . The right syntactic equivalence is the equivalence relation
Similarly, the left syntactic relation is
The syntactic congruence or Myhill congruence may be defined as
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.
Let us call the equivalence class of for the syntactic congruence.
The syntactic congruence is compatible with concatenation in the monoid, in that one has
for all. Thus, the syntactic quotient is a monoid morphism, and induces a quotient monoid
This monoid is called the syntactic monoid of S.
It can be shown that it is the smallest monoid that recognizes S; that is, M recognizes S, and for every monoid N recognizing S, M is a quotient of a submonoid of N. The syntactic monoid of S is also the transition monoid of the minimal automaton of S.
Similarly, a language L is regular if and only if the family of quotients
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. Thus, the number of elements in is at most equal to the number of states of the automaton and is at most the number of final states. Assume, conversely, that the number of elements in is finite. One can then construct an automaton where is the set of states, is the set of final states, the language L is the initial state, and the transition function is given by. Clearly, this automaton recognizes L. Thus, a language L is recognizable if and only if the set is finite. Note that this proof also builds the minimal automaton.
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.

Examples