Green's relations and partial orders on semigroups of partial linear transformations with restricted range

Abstract

Let V be any vector space and P(V) the set of all partial linear transformations defined on V, that is, all linear transformations α: S → T where S; T are subspaces of V. Then P(V) is a semigroup under composition. Let W be a subspace of V. We define PT(V;W) = {α ∈ P(V): Vα ⊆ W}. So PT(V,W) is a subsemigroup of P(V). In this paper, we present the largest regular subsemigroup and determine Green's relations on PT(V;W). Furthermore, we study the natural partial order ≤ on PT(V;W) in terms of domains and images and find elements of PT(V,W) which are compatible. © 2014 by the Mathematical Association of Thailand. All rights reserved.