Almost injective transformations of an infinite set

Abstract

Let V be an infinite-dimensional vector space and let n be a cardinal such that aleph_0<=n<=dim V, and let AM(V,n) denote the semigroup consisting of all linear transformations of V whose nullity is less than n. In recent work, Mendes-Gonçalves and Sullivan studied the ideal structure of AM(V,n). Here, we do the same for a similarly-defined semigroup AM(X,q) of transformations defined on an infinite set X. Although our results are clearly comparable with those already obtained for AM(V,n), we show that the two semigroups are never isomorphic.