Recognizable set
title: "Recognizable set" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["automata-(computation)"] topic_path: "general/automata-computation" source: "https://en.wikipedia.org/wiki/Recognizable_set" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
In computer science, more precisely in automata theory, a recognizable set of a monoid is a subset that can be distinguished by some homomorphism to a finite monoid. 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 subset S\subseteq N is recognized by a monoid M if there exists a homomorphism \phi from N to M such that S=\phi^{-1}(\phi(S)), and recognizable if it is recognized by some finite monoid. 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 \setminus S is in M \setminus T.
Example
Let A be an alphabet: the set A^* of words over A is a monoid, the free monoid on A. The recognizable subsets of A^* are precisely the regular languages. Indeed, such a language is recognized by the transition monoid of any automaton that recognizes the language.
The recognizable subsets of \mathbb N are the ultimately periodic sets of integers.
Properties
A subset of N is recognizable if and only if its syntactic monoid is finite.
The set \mathrm{REC}(N) of recognizable subsets of N is closed under:
Mezei's theorem states that if M is the product of the monoids M_1, \dots, M_n, then a subset of M is recognizable if and only if it is a finite union of subsets of the form R_1 \times \cdots \times R_n, where each R_i is a recognizable subset of M_i. For instance, the subset {1} of \mathbb N is rational and hence recognizable, since \mathbb N is a free monoid. It follows that the subset S={(1,1)} of \mathbb N^2 is recognizable.
McKnight's theorem states that if N is finitely generated then its recognizable subsets are rational subsets. This is not true in general, since the whole N is always recognizable but it is not rational if N is infinitely generated.
Conversely, a rational subset may not be recognizable, even if N is finitely generated. In fact, even a finite subset of N is not necessarily recognizable. For instance, the set {0} is not a recognizable subset of (\mathbb Z, +). Indeed, if a homomorphism \phi from \mathbb Z to M satisfies {0}=\phi^{-1}(\phi({0})), then \phi is an injective function; hence M is infinite.
Also, in general, \mathrm{REC}(N) is not closed under Kleene star. For instance, the set S={(1,1)} is a recognizable subset of \mathbb N^2, but S^*={(n, n)\mid n\in \mathbb N} is not recognizable. Indeed, its syntactic monoid is infinite.
The intersection of a rational subset and of a recognizable subset is rational.
Recognizable sets are closed under inverse of homomorphisms. I.e. if N and M are monoids and \phi:N\rightarrow M is a homomorphism then if S\in\mathrm{REC}(M) then \phi^{-1}(S)={x\mid \phi(x)\in S}\in\mathrm{REC}(N) .
For finite groups the following result of Anissimov and Seifert is well known: a subgroup H of a finitely generated group G is recognizable if and only if H has finite index in G. In contrast, H is rational if and only if H is finitely generated.
References
- Jean-Éric Pin, Mathematical Foundations of Automata Theory, Chapter IV: Recognisable and rational sets
References
- John Meakin. (2007). "Groups St Andrews 2005 Volume 2". Cambridge University Press.
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