BQ-Semigroups of Generalised Linear Transformations

Abstract

If V and W are vector spaces over the same field, we let T(V,W) denote the set of all linear transformations from V into W. In addition, if θ ∈ T(W,V), we define a “sandwich operation” ∗ on T(V,W) by α ∗ β=α θ β for all α, β ∈ T(V,W). Then (T(V,W),∗) is a semigroup of so-called generalised linear transformations, which we denote by T(V,W,θ). A simple result for abstract semigroups shows that T(V,W,θ) belongs to the class BQ of all semigroups whose sets of bi-ideals and quasi-ideals coincide. In recent work, Mendes-Gonçalves and Sullivan examined the same problem for subsemigroups S of T(V,V, id V) for which the dimension (or codimension) of the kernel (or the range) of each α ∈ S is bounded by a fixed cardinal. Here we extend that work to certain subsemigroups of T(V,W,θ) where V ≠ W.