In mathematics and computer science, the syntactic monoid of a formal language is the minimal monoid that recognizes. By the Myhill–Nerode theorem, the syntactic monoid is unique up to unique isomorphism.

Syntactic quotient

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, one may define sets that consist of formal left or right inverses of elements. These are called quotients, and one may define right or left quotients, depending on which side one is concatenating. Thus, the right quotient is the set

Similarly, the left quotient is

Syntactic equivalence

The syntactic quotient induces an equivalence relation, called the syntactic relation, or syntactic equivalence).

The right syntactic equivalence is the equivalence relation

1. .

Similarly, the left syntactic equivalence is

1. .

Observe that the right syntactic equivalence is a left congruence with respect to string concatenation.

The syntactic congruence or Myhill congruence is defined as

1. .

. A disjunctive set is the equality relation.

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

. Thus, the syntactic quotient is a monoid morphism, and induces a quotient monoid

1. .

is called the syntactic monoid. It can be shown that it is the smallest monoid that recognizes is a quotient of a submonoid is also the transition monoid of the minimal automaton.

A group language is one for which the syntactic monoid is a group.

Examples

• .

• , the minimal automaton has 4 states and the syntactic monoid has 15 elements.

• The bicyclic monoid is the syntactic monoid of the Dyck language (the language of balanced sets of parentheses).

• The free monoid, one can use the language of square powers of the letter.)

• Every non-trivial finite monoid is homomorphic to the syntactic monoid of some non-trivial language, but not every finite monoid is isomorphic to a syntactic monoid.

• Every finite group is isomorphic to the syntactic monoid of some regular language.

• .

• Trace monoids are examples of syntactic monoids.

• Marcel-Paul Schützenberger characterized star-free languages as those with finite aperiodic syntactic monoids.

References