Mean-periodic function

In computer science, more precisely in automata theory, a rational set of a monoid is an element of the minimal class of subsets of this monoid which contains all finite subsets and is closed under union, product and Kleene star. Rational sets are useful in automata theory, formal languages and algebra.

Definition

Let $N$ be a monoid. The set $\mathrm{R}\mathrm{A}\mathrm{T}(N)$ of rational subsets of $N$ is the smallest set that contains every finite set and is closed under

• union: if $A,B\in \mathrm{R}\mathrm{A}\mathrm{T}(N)$ then $A\cup B\in \mathrm{R}\mathrm{A}\mathrm{T}(N)$
• product: if $A,B\in \mathrm{R}\mathrm{A}\mathrm{T}(N)$ then $A\times B=\{a\times b\mid a\in A,b\in B\}\in \mathrm{R}\mathrm{A}\mathrm{T}(N)$
• Kleene star if $A\in \mathrm{R}\mathrm{A}\mathrm{T}(N)$ then $A^{*}=\bigcup _{i=1}^{\mathrm{\infty }}{A}^i\in \mathrm{R}\mathrm{A}\mathrm{T}(N)$ where $A^1=A$ and $A^{n+1}=A^n\times A$.

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 the rational subsets of a monoid do not form a submonoid of this monoid.

Example

Let $A$ be an alphabet, the set $A^{*}$ of words over $A$ is a monoid. The rational subset of $A^{*}$ are precisely the regular languages. Indeed this language may be defined by a finite regular expression.

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

Property

McKnight's theorem states that if $N$ is finitely generated then its recognizable subset are rational sets. This is not true in general, i.e. $\mathrm{R}\mathrm{A}\mathrm{T}(N)$ is not closed under complement. Let $N=\{a{\}}^{*}\times \{b,c{\}}^{*}$, the sets $U=\{(a^n,b^n{c}^m)\mid n,m\in \mathbb{\mathbb{N}}\}$ and $V=\{(a^n,b^m{c}^n)\mid n,m\in \mathbb{\mathbb{N}}\}$ are recognizable but $W=U\cap V=\{(a^n,b^n{c}^n)\mid n\in \mathbb{\mathbb{N}}\}$ is not because its projection to the second element $\{b^n{m}^n\mid n\in \mathbb{\mathbb{N}}\}$ is not rational.

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

Rational sets are closed under morphism: given $N$ and $M$ two monoids and $\varphi :N\to M$ a morphism, if $S\in \mathrm{R}\mathrm{A}\mathrm{T}(N)$ then $\varphi (S)=\{\varphi (x)\mid x\in S\}\in \mathrm{R}\mathrm{A}\mathrm{T}(M)$.

See also

• Recognizable 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
• Rational Sets in Commutative Monoids

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