In mathematics and computer science, the syntactic monoid ${\displaystyle M(L)}$ of a formal language ${\displaystyle L}$ is the minimal monoid that recognizes the language ${\displaystyle L}$. By the Myhill–Nerode theorem, the syntactic monoid is unique up to unique isomorphism.

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 ${\displaystyle S}$ of a free monoid ${\displaystyle M}$, one may define sets that consist of formal left or right inverses of elements in ${\displaystyle S}$. These are called quotients, and one may define right or left quotients, depending on which side one is concatenating. Thus, the right quotient of ${\displaystyle S}$ by an element ${\displaystyle m}$ from ${\displaystyle M}$ is the set

Similarly, the left quotient is

The syntactic quotient induces an equivalence relation on ${\displaystyle M}$, called the syntactic relation, or syntactic equivalence (induced by ${\displaystyle S}$).

The right syntactic equivalence is the equivalence relation

Similarly, the left syntactic equivalence is