BQ-semigroups of generalised linear transformations

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 ε T(W,V), we define a "sandwich operation" * on T(V,W) by α * β=α θ β for all α, β ε 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,idV) for which the dimension (or codimension) of the kernel (or the range) of each α ε S is bounded by a fixed cardinal. Here we extend that work to certain subsemigroups of T(V,W,θ) where V ≠ W. © 2013 Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and Suzhou University.