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 alpha in T(V ) for which dim ker >= q and the semigroup OE(p, q) of all alpha in T(V ) for which codim ran >= q, where dim V = p >= q >= aleph0. It is not difficult to see that OM(p, q) and OE(p, q) are a right 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 the Green’s relations and the two-sided ideals of each semigroup, and we determine its maximal regular subsemigroup. Finally, we determine some maximal right cancellative subsemigroups of OE(p, q).