The technique of
covers is now well established in semigroup theory. The idea is, given a semigroup
S, to find a semigroup
having a better understood structure than that of
S, and an onto morphism
of a specific kind from
to
S. With the right conditions on
, the behaviour of
S is closely linked to that of
. If
S is finite one aims to choose a finite
. 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
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
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
to the identity map in the domain of
. In the final part of this paper we consider representations of weakly left quasi-ample semigroups.
Keywords weakly left quasi-ample semigroup - proper cover
This work was supported by the London Mathematical Society, Fundação para a Ciência e a Tecnologia, Centro de Álgebra da Universidade de Lisboa, Centro de Matemática da Universidade do Porto and projects POCTI/32440/MAT/2000 and POCTI/32817/Mat/2000 of FEDER.
Special issue of Studia Logica: 
Algebraic Theory of Quasivarieties
Presented by
M. E. Adams, K. V. Adaricheva, W. Dziobiak, and A. V. Kravchenko