Decidability of the Representation Extension Property for Finite Semigroups

We prove that the decision problem of whether or not a finite semigroup has the representation extension property is decidable. 1. The main theorem and preliminaries It is an immediate consequence of the normal form theorem for amalgamated free products of groups that every amalgam of groups embeds in some group. However, this result fails for semigroup amalgams: an early result of Kimura [6] shows that amalgams of semigroups do not necessarily embed in any semigroup (see also [4], Vol. II, page 138). More recently, Sapir [7] has shown that it is in fact undecidable whether an amalgam of (finite) semigroups embeds in any (finite) semigroup. A semigroup S is called an amalgamation base for semigroups if every amalgam of semigroups containing S as a subsemigroup embeds in some semigroup. It is natural to ask if it is decidable whether or not a finite semigroup is an amalgamation base. According to [3], we say that a semigroup S has the representation extension property if for any right S-set XS and any left S-set SM containing S as a left Ssubset, the canonical map: X → X⊗SM (x 7−→ x⊗1) is injective. Hall [5] proved that any semigroup which is an amalgamation base in the class of all semigroups has the representation extension property. In this paper we prove The Main Theorem. It is decidable whether or not a finite semigroup has the representation extension property. Let S be a semigroup. Let M be a nonempty set with a unitary and associative operation of S : S ×M −→ M((s, w) 7−→ sw), where S is the monoid obtained from S by adjoining a new identity 1. Then M is called a left S-set. Dually, a right S-set is defined. If a left S-set [resp. right S-set] M contains elements m1, · · · ,mn such that M = Sm1 ∪ · · · ∪ Smn [resp. M = m1S ∪ · · · ∪mnS], then we say that m1, · · · ,mn are generators of M . A relation ρ on a left [resp. right] S-set M is called an S-congruence if (m,m′) ∈ ρ and s ∈ S implies (sm, sm′) ∈ ρ [resp. (ms,m′s) ∈ ρ]. Let M,N be left [resp. Received by the editors July 1, 1998. 1991 Mathematics Subject Classification. Primary 20M10.