Some results concerning ample semigroups

A semigroup S is right ample if it can be embedded in an inverse semigroup T such that for all a∊S the idempotent a⁻¹a belongs to S. By a similar token, S is called left ample if aa⁻¹∊S for all a∊S. An ample semigroup is one that is both right and left ample. If S is made left ample and right ample by different inverse semigroups, then the problem of finding a single inverse semigroup T embedding S as a right as well as left ample subsemigroup is, in general, undecidable.
In this talk, we shall consider the above problem when (i) S is finite and (ii) when the containing inverse semigroups are Brandt semigroups. We shall show that, in these cases, the above problem is trivial.