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

Kritsada Sangkhanan, Jintana Sanwong

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 $\alpha:S\to T$ where $S,T$ are subspaces of $V$. Then $P(V)$ is a semigroup under composition. Let $W$ be a subspace of $V$. We define $PT(V,W)=\{\alpha\in P(V):V\alpha\subseteq W\}$. 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 $\leq$ on $PT(V,W)$ in terms of domains and images and find elements of $PT(V,W)$ which are compatible.