Recognizable Set

A recognizable set in a Monoid $M$ is a subset $S\subseteq M$ that can be distinguished by a Monoid Homomorphism to a Finite Monoid.

Explicitly, $S$ is recognizable if there is some Monoid Homomorphism $\varphi:M\to N$ to a finite monoid $N$ such that the Preimage of $\phi(S)$ is $S$.

Example

• A recognizable set of a Free Monoid $\Sigma^{*}$ is precisely a Regular Language over $\Sigma$.