Partial orders on semigroups of partial transformations with restricted range

Abstract

Let X be any set and P(X) the set of all partial transformations defined on X, that is, all functions α:A→B where A,B are subsets of X. Then P(X) is a semigroup under composition. Let Y be a subset of X. Recently, Fernandes and Sanwong defined PT(X,Y)={α P(X):XαY } and defined I(X,Y) to be the set of all injective transformations in PT(X,Y). Hence PT(X,Y) and I(X,Y) are subsemigroups of P(X). In this paper, we study properties of the so-called natural partial order ≥ on PT(X,Y) and I(X,Y) in terms of domains, images and kernels, compare ≥ with the subset order, characterise the meet and join of these two orders, then find elements of PT(X,Y) and I(X,Y) which are compatible. Also, the minimal and maximal elements are described. © 2012 Australian Mathematical Publishing Association Inc.