Green's Relations and Partial Orders on Semigroups of Partial Linear Transformations with Restricted Range

Let V be any vector space and P(V) the set of all partial linear transformations defined on V, that is, all linear transformations alpha : 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) = {alpha is an element of P(V) : V alpha subset of 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.