A Note on Finitely Generated Semigroups of Regular Languages∗
نویسندگان
چکیده
Let E = {E1, . . . , Ek} 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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