Partial Orders on Transformation Semigroups

Abstract

In 1986, Kowol and Mitsch studied properties of the so-called ‘natural partial order’≤ 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), ≤). In this paper, we extend that work to the semigroup P(X) of all partial transformations of X, compare ≤ 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.