Semigroups of Injective Linear Transformations with Infinite Defect
Suzana Mendes-Gonçalves · Communications in Algebra · 2006
Given an infinite-dimensional vector space V, we consider the semigroup KN(p, q) consisting of all injective linear transformations α : V → V, for which the codimension of the range of α is at least q, where dim V = p ≥ q ≥ ℵ0. Kemprasit and Namnak (2001 Kemprasit , Y. , Namnak , C. ( 2001 ). On semigroups of linear transformations whose bi-ideals are quasi-ideals . Pure Math. Appl. 12 ( 4 ): 405 – 413 [CSA] [Google Scholar]) considered the semigroup KN(p, ℵ0) while deciding when certain subsemigroups of T(V )—the semigroup under composition of all linear transformations from V to V—belong to BQ—the class of all semigroups whose sets of bi-ideals and quasi-ideals coincide. In this article, we determine when KN(p, q) belongs to BQ in terms of the dimension of V. Next we characterize 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. Communicated by D. Easdown.