Green’s relations, regularity and abundancy for semigroups of quasi-onto transformations

Abstract

Let $V$ be an infinite-dimensional vector space and for every infinite cardinal $n$ such that $n\le \dim V$, let $AE(V,n)$ denote the semigroup of all linear transformations of $V$ whose defect is less than $n$. In 2009, Mendes-Gonçalves and Sullivan studied the ideal structure of $AE(V,n)$. Here, we consider a similarly-defined semigroup $AE(X,q)$ of transformations defined on an infinite set $X$. Quite surprisingly, the results obtained for sets differ substantially from the results obtained in the linear setting.