Semigroups of injective partial linear transformations with restricted range: Green's relations and partial orders

Abstract

Let V be any vector space and I(V) the set of all partial injective linear transformations defined on V, that is, all injective linear transformations α: A → B where A, B are subspaces of V. Then I(V) is a semigroup under composition. Let W be a subspace of V. Define I(V, W)={α∈ I(V): V α ⊆ W}. So I(V,W) is a subsemigroup of I(V). In this paper, we present the largest regular subsemigroup of I(V, W) and determine its Green's relations. Furthermore, we study the natural partial order ≤ on I(V, W) in terms of domains and images, compare ≤ with the subset order and find elements of I(V, W) which are compatible. © 2012 Academic Publications, Ltd.