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 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) 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.

Key Words:

• Bi-ideals
• Quasi-ideals
• Semigroups of linear transformations

AMS Classification:

• Primary 20M20
• Secondary 15A04

ACKNOWLEDGMENTS

The author would like to thank Professor R. P. Sullivan for the subject, and to express her gratitude for his valuable comments and corrections of the original draft.

The author acknowledges the support of the Portuguese Foundation for Science and Technology (FCT) through the research program POCTI.