Baer-Levi semigroups of linear transformations

Abstract

Given an infinite-dimensional vector space V, we consider the semigroup GS(m,n) of all injective linear transformations of V into itself with defect n, where n is an infinite cardinal less or equal than m, the dimension of V. This is a linear version of the well-known Baer-Levi semigroup BL(p,q) defined on an infinite set X with cardinal p and where q is an infinite cardinal less or equal than p. We show that, although the basic properties of GS(m,n) are the same as those of BL(p,q), the two semigroups are never isomorphic. We also determine all left ideals of GS(m,n) and some of its maximal subsemigroups: in this, we follow previous work on BL(p,q) by Sutov (1966) and Sullivan (1978) as well as Levi and Wood (1984).