Partial orders on transformation semigroups

Abstract

In 1986, Kowol and Mitsch studied properties of the so-called 'natural partial order' less than or equal to on T(X), the total transformation semigroup defined on a set X. In particular, they determined when two total transformations are related under this order, and they described the minimal and maximal elements of (T(X), less than or equal to). In this paper, we extend that work to the semigroup P(X) of all partial transformations of X, compare less than or equal to with another 'natural' partial order on P(X), characterise the meet and join of these two orders, and determine the minimal and maximal elements of P(X) with respect to each order.