A Note on Finitely Generated Semigroups of Regular Languages

Abstract

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.