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

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 , called the syntactic relation, or syntactic equivalence  (induced by ).

The right syntactic equivalence is the equivalence relation

.

Similarly, the left syntactic equivalence is

.

Observe that the right syntactic equivalence is a left congruence with respect to string concatenation and vice versa; i.e.,  for all .

The syntactic congruence or Myhill congruence is defined as

.

The definition extends to a congruence defined by a subset  of a general monoid .  A disjunctive set is a subset  such that the syntactic congruence defined by  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 .
It can be shown that it is the smallest monoid that recognizes ; that is,  recognizes , and for every monoid  recognizing ,  is a quotient of a submonoid of . The syntactic monoid of  is also the transition monoid of the minimal automaton of .

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

Myhill–Nerode theorem

The Myhill–Nerode theorem states: a language  is regular if and only if the family of quotients  is finite, or equivalently, the left syntactic equivalence  has finite index (meaning it partitions  into finitely many equivalence classes).

This theorem was first proved by Anil Nerode  and the relation  is thus referred to as Nerode congruence by some authors.

Proof

The proof of the "only if" part is as follows. Assume that a finite automaton recognizing  reads input  which leads to state .  If  is another string read by the machine, also terminating in the same state , 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.

For a proof of the "if" part, assume 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  is the initial state, and the transition function is given by . Clearly, this automaton recognizes . Thus, a language  is recognizable if and only if the set  is finite. Note that this proof also builds the minimal automaton.

Examples
 Let  be the language over  of words of even length.  The syntactic congruence has two classes,  itself and , the words of odd length.  The syntactic monoid is the group of order 2 on .  
 The bicyclic monoid is the syntactic monoid of the Dyck language (the language of balanced sets of parentheses).
 The free monoid on  (where ) is the syntactic monoid of the language , where  is the reversal of the word .
 Every 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 non-trivial language.
 The language over  in which the number of occurrences of  and  are congruent modulo  is a group language with syntactic monoid .
 Trace monoids are examples of syntactic monoids.
 Marcel-Paul Schützenberger characterized star-free languages as those with finite aperiodic syntactic monoids.

References

 
 
 
 
 
 

Formal languages
Semigroup theory