Partial orders on linear transformation semigroups

Let V be any vector space and P(V) the set of all partial linear transformations defined on V, that is, all linear alpha : A -> B, where A, B are subspaces of V. Then P(V) is a semigroup under composition, which is partially ordered by C (that is, alpha subset of beta if and only if dom alpha subset of dom beta and alpha = beta vertical bar dom alpha). We compare this order with the so-called 'natural partial order' <= on P(V) and we determine their meet and join. We also describe all elements of P(V) that are minimal (or maximal) with respect to each of these four orders, and we characterize all elements that are 'compatible' with them. In addition, we answer similar questions for the semigroup T(V) consisting of all alpha epsilon P(V) whose domain equals V. Other orders have been defined by Petrich on any regular semigroup: three of them form a chain below <=(1) and we show that two of these are equal on the semigroup P(V) and on the ring T(V). We also consider questions for these orders that are similar to those already mentioned.