An application of a Theorem of Ash to finite covers

Abstract

The technique of covers is now well established in semigroup theory. The idea is, given a semigroup S, to find a semigroup Ŝ having a better understood structure than that of S, and an onto morphism θ of a specific kind from Ŝ to S. With the right conditions on θ, the behaviour of S is closely linked to that of Ŝ. If S is finite one aims to choose a finite Ŝ. The celebrated results for inverse semigroups of McAlister in the 1970’s form the flagship of this theory.

Weakly left quasi-ample semigroups form a quasivariety (of algebras of type(2, 1)), properly containing the classes of groups, and of inverse, left ample, and weakly left ample semigroups. We show how the existence of finite proper covers for semigroups in this quasivariety is a consequence of Ash’s powerful theorem for pointlike sets. Our approach is to obtain a cover Ŝ of a weakly left quasi-ample semigroup S as a subalgebra of S × G, where G is a group. It follows immediately from the fact that weakly left quasi-ample semigroups form a quasivariety, that Ŝ is weakly left quasi-ample. We can then specialise our covering results to the quasivarieties of weakly left ample, and left ample semigroups. The latter have natural representations as (2, 1)-subalgebras of partial (one-one) transformations, where the unary operation takes a transformation α to the identity map in the domain of α. In the final part of this paper we consider representations of weakly left quasi-ample semigroups.