Rees matrix covers for regular semigroups

Abstract

By a local submonoid, of a regular semigroup S, we mean a subset of the form eSe, where e is an idempotent of S. Many classes of regular semigroups can be defined in terms of properties of their local submonoids. For example, rectangular bands can be characterized as those regular semigroups all of whose local submonoids are trivial while completely simple semigroups are those whose local submonoids are groups. We say that a regular semigroup has a property ‘3 locally, or is a locally 5?7 semigroup, if each local submonoid of S has property %Y. In a previous paper we showed that a regular semigroup is locally inverse if and only if it is an image, by a homomorphism which is one to one on local submonoids, of a regular Rees matrix semigroup over an inverse semigroup. In this paper we extend that result to various other classes of regular semigroups. In particular, we show the analog of this result for locally E-solid semigroups. (A regular semigroup is E-solid if the subsemigroup generated by its idempotents is a union of groups.) The class of locally E-solid regular semigroups is extremely extensive. It includes almost all classes of regular semigroups which have been studied from a structural point of view since inverse semigroups, orthodox semigroups, unions of groups semigroups and their localizations all belong to this class. The first section of the paper contains preliminary results which we shall require later in the paper. Section 2 describes a general procedure for constructing Rees matrix covers for regular semigroups which is applied in subsequent sections to obtain Rees matrix covers for special classes of regular semigroups. These classes include locally E-solid regular semigroups, locally orthodox semigroups and locally Sunipotent semigroups. The covering theorems on these classes of regular semigroups can be obtained directly through a painstaking analaysis of special cases. We have, however, chosen to obtain them as applications of the covering theorem for locally inverse semigroups which was proved earlier [ 81 by making use of recent interesting results of Hall concerning the relationship between locally E-solid