Regular elements and Green's relations in Generalised Linear Transformation Semigroups

Abstract

If V and W are vector spaces over the same field, we let P(V,W) denote the set of all partial linear transformations from V into W (that is, all linear mappings whose domain and range are subspaces of V and W, respectively). If $\theta\in P(W,V)$, then P(V,W) is a so-called `generalised semigroup' of linear transformations under the `sandwich operation': $\alpha *\beta=\alpha\circ\theta\circ\beta$, for each $\alpha,\beta\in P(V,W)$. We denote this semigroup by $P(V,W,\theta)$ and, in this paper, we characterise Green's relations on it: that is, we study equivalence relations which determine when principal left (or right, or 2-sided) ideals in $P(V,W,\theta)$ are equal. This is related to a problem raised by Magill and Subbiah in 1975. We also discuss the same idea for important subsemigroups of $P(V,W,\theta)$ and characterise when these semigroups satisfy certain regularity conditions.