The Ideal Structure of Semigroups of Linear Transformations with Lower Bounds on Their Nullity or Defect

Abstract

Suppose V is an infinite-dimensional vector space and let T(V) denote the semigroup (under composition) of all linear transformations of V. In this paper, we study the semigroup OM(p,q) consisting of all α ∈ T(V) for which dim ker α ≥ q and the semigroup OE(p,q) of all α ∈ T(V) for which codim ran α ≥ q, where dim V = p ≥ q ≥ ℵ0. It is not difficult to see that OM(p,q) and OE(p,q) are a right ideal and a left ideal of T(V), respectively, and using these facts, we show that they belong to the class of all semigroups whose sets of bi-ideals and quasi-ideals coincide. Also, we describe Green's relations and the two-sided ideals of each semigroup, and determine its maximal regular subsemigroup. Finally, we determine some maximal right cancellative subsemigroups of OE(p,q).