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

Abstract

Suppose $V$ is a vector space with ${\rm dim} V=p\geq q\geq\aleph_0$, and let $T(V)$ denote the semigroup (under composition) of all linear transformations of $V$. For each $\alpha\in T(V)$, let ${\rm ker}\alpha$ and ${\rm ran}\alpha$ denote the `kernel' and the `range' of $\alpha$, and write $n(\alpha)={\rm dim}{\rm ker}\alpha$ and $d(\alpha)={\rm codim}{\rm ran}\alpha$. In this paper, we study the semigroups $AM(p,q) =\{\alpha\in T(V):n(\alpha)<q\}$ and $AE(p,q) =\{\alpha\in T(V):d(\alpha)<q\}$. 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 $AM(p,q)$.