Inverse semigroups generated by linear transformations

Abstract

Suppose X is a set with cardinal p and let q be an infinite cardinal less or equal than p. Let B=BL(p,q) denote the Baer-Levi semigroup defined on X. In 1984, Howie and Marques-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^{-1}B. 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.