Rational set

Incomputer science,more precisely inautomata theory,arational setof amonoidis an element of the minimal class of subsets of this monoid that contains all finite subsets and is closed underunion,product andKleene star.Rational sets are useful inautomata theory,formal languagesandalgebra.

A rational set generalizes the notion ofrational (or regular) language(understood as defined byregular expressions) to monoids that are not necessarilyfree.^{[example needed]}

Definition

Let $(N,\cdot )$ be amonoidwith identity element $e$ .The set $\mathrm {RAT} (N)$ of rational subsets of $N$ is the smallest set that contains every finite set and is closed under

union:if $A,B\in \mathrm {RAT} (N)$ then $A\cup B\in \mathrm {RAT} (N)$
product: if $A,B\in \mathrm {RAT} (N)$ then $A\cdot B=\{a\cdot b\mid a\in A,b\in B\}\in \mathrm {RAT} (N)$
Kleene star:if $A\in \mathrm {RAT} (N)$ then $A^{*}=\bigcup _{i=0}^{\infty }A^{i}\in \mathrm {RAT} (N)$ where $A^{0}=\{e\}$ is the singleton containing the identity element, and where $A^{n+1}=A^{n}\cdot A$ .

This means that any rational subset of $N$ can be obtained by taking a finite number of finite subsets of $N$ and applying the union, product and Kleene star operations a finite number of times.

In general a rational subset of a monoid is not a submonoid.

Example

Let $A$ be analphabet,the set $A^{*}$ ofwordsover $A$ is a monoid. The rational subset of $A^{*}$ are precisely theregular languages.Indeed, the regular languages may be defined by a finiteregular expression.

The rational subsets of $\mathbb {N}$ are the ultimately periodic sets of integers. More generally, the rational subsets of $\mathbb {N} ^{k}$ are thesemilinear sets.^[1]

Properties

McKnight's theoremstates that if $N$ isfinitely generatedthen itsrecognizable subsetare rational sets. This is not true in general, since the whole $N$ is always recognizable but it is not rational if $N$ is infinitely generated.

Rational sets are closed under homomorphism: given $N$ and $M$ two monoids and $\phi:N\rightarrow M$ a monoid homomorphism, if $S\in \mathrm {RAT} (N)$ then $\phi (S)=\{\phi (x)\mid x\in S\}\in \mathrm {RAT} (M)$ .

$\mathrm {RAT} (N)$ is not closed undercomplementas the following example shows.^[2] Let $N=\{a\}^{*}\times \{b,c\}^{*}$ ,the sets $R=(a,b)^{*}(1,c)^{*}=\{(a^{n},b^{n}c^{m})\mid n,m\in \mathbb {N} \}$ and $S=(1,b)^{*}(a,c)^{*}=\{(a^{n},b^{m}c^{n})\mid n,m\in \mathbb {N} \}$ are rational but $R\cap S=\{(a^{n},b^{n}c^{n})\mid n\in \mathbb {N} \}$ is not because its projection to the second element $\{b^{n}c^{n}\mid n\in \mathbb {N} \}$ is not rational.

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

Forfinite groupsthe following result of A. Anissimov and A. W. Seifert is well known: asubgroupHof afinitely generated groupGis recognizable if and only ifHhasfinite indexinG.In contrast,His rational if and only ifHis finitely generated.^[3]

Rational relations and rational functions

Abinary relationbetween monoidsMandNis arational relationif the graph of the relation, regarded as a subset ofM×Nis a rational set in the product monoid. A function fromMtoNis arational functionif the graph of the function is a rational set.^[4]

References

Diekert, Volker; Kufleitner, Manfred; Rosenberg, Gerhard; Hertrampf, Ulrich (2016). "Chapter 7: Automata".Discrete Algebraic Methods.Berlin/Boston: Walter de Gruyther GmbH.ISBN 978-3-11-041332-8.
Jean-Éric Pin,Mathematical Foundations of Automata Theory,Chapter IV: Recognisable and rational sets
Samuel EilenbergandMarcel-Paul Schützenberger,Rational Sets in Commutative Monoids,Journal of Algebra,1969.

^Mathematical Foundations of Automata Theory
^cf. Jean-Éric Pin,Mathematical Foundations of Automata Theory,p. 76, Example 1.3
^John Meakin (2007). "Groups and semigroups: connections and contrasts". In C.M. Campbell; M.R. Quick; E.F. Robertson; G.C. Smith (eds.).Groups St Andrews 2005 Volume 2.Cambridge University Press. p. 376.ISBN 978-0-521-69470-4.preprint
^Hoffmann, Michael; Kuske, Dietrich; Otto, Friedrich; Thomas, Richard M. (2002). "Some relatives of automatic and hyperbolic groups". In Gomes, Gracinda M. S. (ed.).Semigroups, algorithms, automata and languages. Proceedings of workshops held at the International Centre of Mathematics, CIM, Coimbra, Portugal, May, June and July 2001.Singapore: World Scientific. pp. 379–406.Zbl 1031.20047.