Syntactic monoid

Smallest monoid that recognizes a formal language

In mathematics and computer science, the syntactic monoid ${\displaystyle M(L)}$ of a formal language ${\displaystyle L}$ is the smallest monoid that recognizes the language ${\displaystyle L}$.

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 ${\displaystyle S}$ 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

${\displaystyle S\ /\ m=\{u\in M\;\vert \;um\in S\}.}$

Similarly, the left quotient is

${\displaystyle m\setminus S=\{u\in M\;\vert \;mu\in S\}.}$

Syntactic equivalence

The syntactic quotient^{[clarification needed]} 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

${\displaystyle s\sim _{S}t\ \Leftrightarrow \ S\,/\,s\;=\;S\,/\,t\ \Leftrightarrow \ (\forall x\in M\colon \ xs\in S\Leftrightarrow xt\in S)}$.

Similarly, the left syntactic equivalence is

${\displaystyle s\;{}_{S}{\sim }\;t\ \Leftrightarrow \ s\setminus S\;=\;t\setminus S\ \Leftrightarrow \ (\forall y\in M\colon \ sy\in S\Leftrightarrow ty\in S)}$.

Observe that the right syntactic equivalence is a left congruence with respect to string concatenation and vice versa; i.e., ${\displaystyle s\sim _{S}t\ \Rightarrow \ xs\sim _{S}xt\ }$ for all ${\displaystyle x\in M}$.

The syntactic congruence or Myhill congruence^[1] is defined as^[2]

${\displaystyle s\equiv _{S}t\ \Leftrightarrow \ (\forall x,y\in M\colon \ xsy\in S\Leftrightarrow xty\in S)}$.

The definition extends to a congruence defined by a subset ${\displaystyle S}$ of a general monoid ${\displaystyle M}$. A disjunctive set is a subset ${\displaystyle S}$ such that the syntactic congruence defined by ${\displaystyle S}$ is the equality relation.^[3]

Let us call ${\displaystyle [s]_{S}}$ the equivalence class of ${\displaystyle s}$ for the syntactic congruence. The syntactic congruence is compatible with concatenation in the monoid, in that one has

${\displaystyle [s]_{S}[t]_{S}=[st]_{S}}$

for all ${\displaystyle s,t\in M}$. Thus, the syntactic quotient is a monoid morphism, and induces a quotient monoid

${\displaystyle M(S)=M\ /\ {\equiv _{S}}}$.

This monoid ${\displaystyle M(S)}$ is called the syntactic monoid of ${\displaystyle S}$. It can be shown that it is the smallest monoid that recognizes ${\displaystyle S}$; that is, ${\displaystyle M(S)}$ recognizes ${\displaystyle S}$, and for every monoid ${\displaystyle N}$ recognizing ${\displaystyle S}$, ${\displaystyle M(S)}$ is a quotient of a submonoid of ${\displaystyle N}$. The syntactic monoid of ${\displaystyle S}$ is also the transition monoid of the minimal automaton of ${\displaystyle S}$.^[1][2][4]

A group language is one for which the syntactic monoid is a group.^[5]