Inverse semigroups generated by linear transformations

Suppose X is a set with | X| = p ≥ q ≥ No and let B = BL(p, q) denote the Baer-Levi semigroup defined on X. In 1984, Howie and Maxques-Smith showed that, if p = q, then BB-1 = I(X), the symmetric inverse semigroup, on X, and they described the subsemigroup of I(X) generated by B-1B. In 1994, Lima extended that work to 'independence algebras', and thus also to vector spaces. In this paper, we answer the natural question: what happens when p > q? We also show that, in this case, the analogues BB-1 for sets and GG(-1) for vector spaces are never isomorphic, despite their apparent similarities.