Almost Injective Transformations of an Infinite Set

Abstract

Let V be an infinite-dimensional vector space, let n be a cardinal such that ℵ₀ ≤ n ≤ dim V, and let AM(V, n) denote the semigroup consisting of all linear transformations of V whose nullity is less than n. In recent work, Mendes-Gonçalves and Sullivan studied the ideal structure of AM(V, n). Here, we do the same for a similarly-defined semigroup AM(X, q) of transformations defined on an infinite set X. Although our results are clearly comparable with those already obtained for AM(V, n), we show that the two semigroups are never isomorphic.

Key Words:

• Almost one-to-one transformations
• Green's relations
• Ideals

2000 Mathematics Subject Classification:

• Primary 20M20
• Secondary 15A04

ACKNOWLEDGMENT

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

Notes

Communicated by V. Gould.

*Current e-mail: [email protected]