The set of all regular languages is closed under concatenation and forms a monoid known as the monoid of regular languages. In this paper the structure of finitely generated subsemigroups of this monoid in case of one letter alphabet is investigated. We prove that finitely generated semigroups of regular languages over a one letter alphabet are Kleene, rational and thus automatic. It is already known that not all of finitely generated commutative semigroups are automatic, thus we may conclude that semigroups of unary regular languages have more rigid structure.
Let Σ be a finite alphabet. A set [Formula: see text] of regular languages over Σ is called rational if there exists a finite set [Formula: see text] of regular languages over Σ such that [Formula: see text] is a rational subset of the finitely generated semigroup [Formula: see text] with [Formula: see text] as the set of generators and language concatenation as a product. We prove that for any rational set [Formula: see text] and any regular language R ⊆ Σ* it is decidable (1) whether [Formula: see text] or not, and (2) whether [Formula: see text] is finite or not. Possible applications to semistructured databases query processing are discussed.
Let E = {E 1 ,. .. , E k } be a set of regular languages over a finite alphabet Σ. Consider morphism ϕ : ∆ + → (S, •) where ∆ + is the semigroup over a finite set ∆ and (S, •) = E is the finitely generated semigroup with E as the set of generators and language concatenation as a product. We prove that the membership problem of the semigroup S, the set [u] = {v ∈ ∆ + | ϕ(v) = ϕ(u)}, is a regular language over ∆, while the set Ker(ϕ) = {(u, v) | u, v ∈ ∆ + ϕ(u) = ϕ(v)} need not to be regular. It is conjectured however that every semigroup of regular languages is automatic.
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