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In computer science, more precisely in automata theory, a recognizable set of a monoid is a submonoid which can be mapped in a certain sense to some finite monoid through some morphism. Recognizable sets are useful in automata theory, formal languages and algebra.

This notion is different from the notion of recognizable language. Indeed, the term "recognizable" has a different meaning in computability theory.

Definition

Let $N$ be a monoid, a submonoid $S\subseteq N$ is recognizable if there exists a morphism $\varphi$ from $N$ to a finite monoid $M$ such that $S={\varphi }^{-1}(\varphi (S))$. This means that there exists a subset $T$ of $M$ (not necessarily a submonoid of $M$) such that the image of $S$ is in $T$ and the image of $N/S$ is in $M/T$.

Example

Let $A$ be an alphabet, the set $A^{*}$ of words over $A$ is a monoid. The recognizable subset of $A^{*}$ are precisely the regular languages. Indeed this language is recognized by the transition monoid of any automaton that recognizes the language.

The recognizable subsets of $\mathbb{\mathbb{N}}$ are the ultimately periodic sets of integers.

Property

A subset of $N$ is recognizable if and only if its syntactic monoid is finite.

The set $\mathrm{R}\mathrm{E}\mathrm{C}(N)$ of recognizable subsets of $N$ contains every finite subset of $N$ and is closed under:

• union
• intersection
• complement
• right and left quotient

McKnight's theorem states that if $N$ is finitely generated then its recognizable subsets are rational subsets. This is not true in general, i.e. $\mathrm{R}\mathrm{E}\mathrm{C}(N)$ is not closed under Kleene star. Let $N={\mathbb{\mathbb{N}}}^2$, the set $S=\{(1,1)\}$ is finite, hence recognizable, but $S^{*}=\{(i,i)\mid i\in \mathbb{\mathbb{N}}\}$ is not recognizable. Indeed its syntactic monoid is infinite.

The intersection of a rational subset and of a recognizable subset is rational.

Rational sets are closed under inverse morphism. I.e. if $N$ and $M$ are monoid and $\varphi :N\to M$ is a morphism then if $S\in \mathrm{R}\mathrm{E}\mathrm{C}(M)$ then ${\varphi }^{-1}(S)=\{x\mid \varphi (x)\in S\}\in \mathrm{R}\mathrm{E}\mathrm{C}(M)$.

See also

• Rational set

References

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
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• Mathematical Foundations of Automata Theory