Amalgamating inverse semigroups over ample semigroups

Abstract

We consider semigroup amalgams (S; T1, T2) in which T1 and T2 are inverse semigroups and S is a non-inverse semigroup. They are known to be non-embeddable if T1 and T2 are both groups (Clifford semigroups), but S is not such. We prove that (S; T1, T2) is non-embeddable if S is a non-inverse ample semigroup. By introducing the notion of rich ampleness, we determine some necessary and sufficient conditions for the weak embedding of (S; T1, T2) in an inverse semigroup. In particular, (S; T1, T2) is shown to be weakly embeddable in a group if T1 and T2 are groups. A rudimentary analysis of the novel classes of rich ample semigroups is also provided.