Enlargements of Regular Semigroups

Quasi-ideals were introduced by Otto Steinfeld [43] as those non-empty subsets Q of a semigroup T satisfying QTD TQ c Q. When T is regular they are precisely the subsets Q of T which satisfy QTQ = Q ([43, Theorem 9.3]). There are many examples of quasi-ideals in regular semigroup theory. We list below some of the most important: • Every subsemigroup of the form eSe (where e is an idempotent) is a quasi-ideal of S. Such subsemigroups are called local submonoids. • McAlister [30] studies quasi-ideal embeddings of one regular semigroup in another. • Quasi-ideal embeddings arise naturally in the inverse case. Let I be a structure whose semigroup of partial automorphisms F(Z) is an inverse semigroup. Let E be embedded in another structure Z' of the same type such that the semigroup F(E') of partial automorphisms is also an inverse semigroup. Then F(E) is a quasi-ideal of F(E'). • In [24] and [29], McAlister shows that every inverse semigroup can be embedded as a quasi-ideal in a factorisable inverse semigroup. If S is a regular subsemigroup and quasi-ideal of the regular semigroup T (to be brief, we call S a regular quasi-ideal of T), then the semigroups S and T may still have little in common. Consider now the subset T = TST. It can be shown that 7" is a regular subsemigroup of T containing S. But it is also easily verified that both S = ST'S and T = TST. Thus S is embedded as a quasi-ideal of T possessing the additional property that T = T'ST. We call such embeddings enlargements, and it is the thesis of this paper that enlargements provide a useful new idea in the study of regular semigroups. Two examples may serve to provide some partial support to this contention.