Semigroups of injective linear transformations with infinite defect

Abstract

Given an infinite-dimensional vector space $V$, we consider the semigroup $KN(p,q)$ consisting of all injective linear transformations $\alpha:V\to V$ for which the codimension of the range of $\alpha$ is at least $q$, where $\dim V=p\geq q\geq\aleph_0$. In 2001, Kemprasit and Namnak considered the semigroup $KN(p,\aleph_0)$ while deciding when certain subsemigroups of $T(V)$, the semigroup under composition of all linear transformations from $V$ to $V$, belong to ${\bf BQ}$, the class of all semigroups whose sets of bi-ideals and quasi-ideals coincide. In this paper, we determine when $KN(p,q)$ belongs to ${\bf BQ}$ in terms of the dimension of $V$. Next we characterise Green's relations on $KN(p,q)$ and determine its one and two sided ideals; and we use this information to show that $KN(p,q)$ is a model for certain types of algebraic semigroups. Then we describe all quasi-ideals and bi-ideals of $KN(p,q)$. We also determine its maximal right simple subsemigroups.