An application of a Theorem of Ash to finite covers

Weakly left ample semigroups form a class of semigroups that are (2; 1)-subalgebras of semigroups of partial transformations, where the unary operation takes a transformation to the identity map in the domain of. They are semigroups with commuting idempotents such that the e R-class of any element a contains a (necessarily unique) idempotent a + , e R is a left congruence, and the ample identity ae = (ae) + a is satissed for all elements a and idempotents e. Weakly left ample semigroups form a quasivariety (of algebras of type (2; 1)), properly containing the classes of inverse and left ample semigroups. In an earlier paper we gave an explicit construction of a nite proper cover for a nite weakly left ample semigroup. Dropping the conditions that idempotents commute and that e R is a left congruence, but insisting that the e R-class of any element a contains a unique idempotent a + and the ample identity holds, we obtain the strictly larger quasivariety of weakly left quasi-ample semigroups. We show how the existence of nite proper covers for semigroups in this quasivariety is a consequence of Ash's celebrated theorem for pointlike sets.