ON THE STRUCTURE OF REGULAR SEMIGROUPS IN WHICH THE MAXIMAL SUBGROUPS FORM A BAND OF GROUPS

Abstract

A regular semigroup S is said to be quasi-orthodox if and only if there exist an inverse semigroup I and a surjective homomorphism f: S → I such that ef−1 is a completely simple subsemigroup of S for each idempotent e of I. If a regular semigroup S satisfies the following property P, then S is necessarily quasi-orthodox: (P) The maximal subgroups of S form a band of groups. Such a semigroup S is called a quasi-orthodox semigroup with (P). In this paper, the structure of quasi-orthodox semigroups with (P) is studied. Structure theorems are established for the class of general quasi-orthodox semigroups and for some special classes of quasi-orthodox semigroups. In particular the concept of spined product of orthodox semigroups with (P) is introduced, and it is shown that an orthodox semigroup S is isomorphic to the spined product of an H-degenerated orthodox semigroup and an H-compatible inverse semigroup if and only if S has the property (P).