The ideal structure of semigroups of linear transformations with upper bounds on their nullity or defect

Suppose V is a vector space with dim V = p ≥ q ≥ 0, and let T(V) denote the semigroup (under composition) of all linear transformations of V. For α T (V), let ker α and ran α denote the “kernel” and the “range” of α, and write n(α) = dim ker α and d(α) = codim ran α. In this article, we study the semigroups AM(p, q) = {α T(V):n(α) <q} and AE(p, q) = {α T(V):d(α) <q}. First, we determine whether they belong to the class of all semigroups whose sets of bi-ideals and quasi-ideals coincide. Then, for each semigroup, we describe its maximal regular subsemigroup, and we characterise its Green's relations and (two-sided) ideals. As a precursor to further work in this area,, we also determine all the maximal right simple subsemigroups of AM(p, q).