The regular part of a semigroup of linear transformations with restricted range

Abstract

© 2017 Australian Mathematical Publishing Association Inc. Let V be a vector space and let T(V) denote the semigroup (under composition) of all linear transformations from V into V. For a fixed subspace W of V, let T(V;W) be the semigroup consisting of all linear transformations from V into W. In 2008, Sullivan ['Semigroups of linear transformations with restricted range', Bull. Aust. Math. Soc. 77(3) (2008), 441-453] proved that Q = {α ϵ T(V;W) : Vα ⊆ Wα} is the largest regular subsemigroup of T(V;W) and characterized Green's relations on T(V;W). In this paper, we determine all the maximal regular subsemigroups of Q whenW is a finite-dimensional subspace of V over a finite field. Moreover, we compute the rank and idempotent rank of Q when W is an ndimensional subspace of an m-dimensional vector space V over a finite field F.