Rational set explained
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 that contains all finite subsets and is closed under union, product and Kleene star. Rational sets are useful in automata theory, formal languages and algebra.
A rational set generalizes the notion of rational (or regular) language (understood as defined by regular expressions) to monoids that are not necessarily free.
Definition
Let
be a
monoid with identity element
. The set
of rational subsets of
is the smallest set that contains every finite set and is closed under
then
then
A ⋅ B=\{a ⋅ b\mida\inA,b\inB\}\inRAT(N)
then
where
is the singleton containing the identity element, and where
.
This means that any rational subset of
can be obtained by taking a finite number of finite subsets of
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
be an
alphabet, the set
of words over
is a monoid. The rational subset of
are precisely the
regular languages. Indeed, the regular languages may be defined by a finite
regular expression.
The rational subsets of
are the ultimately periodic sets of integers. More generally, the rational subsets of
are the semilinear sets.
[1] Properties
McKnight's theorem states that if
is finitely generated then its
recognizable subset are rational sets.This is not true in general, since the whole
is always recognizable but it is not rational if
is infinitely generated.
Rational sets are closed under homomorphism: given
and
two monoids and
a monoid homomorphism, if
then
\phi(S)=\{\phi(x)\midx\inS\}\inRAT(M)
.
is not closed under
complement as the following example shows.
[2] Let
, the sets
R=(a,b)*(1,c)*=\{(an,bncm)\midn,m\inN\}
and
S=(1,b)*(a,c)*=\{(an,bmcn)\midn,m\inN\}
are rational but
R\capS=\{(an,bncn)\midn\inN\}
is not because its projection to the second element
is not rational.
The intersection of a rational subset and of a recognizable subset is rational.
For finite groups the following result of A. Anissimov and A. W. 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.[3]
Rational relations and rational functions
A binary relation between monoids M and N is a rational relation if the graph of the relation, regarded as a subset of M×N is a rational set in the product monoid. A function from M to N is a rational function if the graph of the function is a rational set.[4]
See also
References
- Book: Diekert . Volker . Kufleitner . Manfred . Rosenberg . Gerhard . Hertrampf . Ulrich . Discrete Algebraic Methods . 2016 . Walter de Gruyther GmbH . Berlin/Boston . 978-3-11-041332-8 . Chapter 7: Automata.
Further reading
- Book: Sakarovitch, Jacques . Elements of automata theory . Translated from the French by Reuben Thomas . Cambridge . Cambridge University Press . 2009 . 978-0-521-84425-3 . 1188.68177 . Part II: The power of algebra .
Notes and References
- http://www.liafa.jussieu.fr/~jep/MPRI/MPRI.html Mathematical Foundations of Automata Theory
- cf. Jean-Éric Pin, Mathematical Foundations of Automata Theory, p. 76, Example 1.3
- Book: C.M. Campbell . M.R. Quick . E.F. Robertson . G.C. Smith. Groups St Andrews 2005 Volume 2. 2007. Cambridge University Press. 978-0-521-69470-4. 376. Groups and semigroups: connections and contrasts . John Meakin. preprint
- Book: Hoffmann . Michael . Kuske . Dietrich . Otto . Friedrich . Thomas . Richard M. . Some relatives of automatic and hyperbolic groups . 1031.20047 . Gomes . Gracinda M. S. . 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 . 379–406 . 2002 .
