Inverse Semigroups and Extensions of Groups by Semilattices

This paper is the first part of a series of three papers devoted to the study of inverse semigroups. The subject of our second paper [7] is free inverse semigroups, the third one [S] is dedicated to finite inverse semigroups and applications to language theory, while this one is concerned with general inverse semigroups. Much of the structure theory of inverse semigroups has revolved about constructing an arbitrary inverse semigroup from groups and semilattices, and the main results of this theory can be stated as follows. An Esemigroup S (that is, a semigroup whose idempotents commute) is said to be an extension of a group by a semilattice if there is a surjective morphism 4 from S onto a group such that 14 ~ ’ is the set of idempotents of S. First, every inverse semigroup is covered by a regular extension of a group by a semilattice and the covering map is one-to-one on idempotents. Second, regular extensions of groups by semilattices are exactly E-unitary inverse semigroups [17], or P-semigroups (in the sense of McAlister [9, lo]), or regular subsemigroups of semidirect products of a semilattice by a group C161. The aim of this paper is to develop a similar theory in the non-regular case. However, as usual in semigroup theory, many difficulties arise when passing from the regular case to the non-regular case. The first obvious problem is to find non-regular equivalents to the notions of inverse semigroups, E-unitary inverse semigroups, P-semigroups, etc. It turns out