Extensions of a Brandt Semigroup by Another

Abstract

One possible first step in considering the structure of a class of semigroups is to study the ideal extensions (here simply called “extensions“) of simple or 0-simple semigroups in by another if the latter are of known structure. Extensions of a semigroup by another were first studied by Clifford (see [1, 4.4 and 4.5]). In his constructions, an extension of a semigroup S by a semigroup T with zero is given by a function (satisfying certain conditions) from T* = T\0 into the translational hull of S.We use certain results (refining those of Clifford) established in [2] and a description of the translational hull of a Brandt semigroup given in [9] (see also [8]), to construct all extensions V of a Brandt semigroup S having a finite number of idempotents by any Brandt semigroup T (cf. [10]).