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
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
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
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 .
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).
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.
- 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[clarification needed] 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.
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