Partial orders on partial Baer-Levi semigroups

Abstract

Marques-Smith and Sullivan [Partial orders on transformation semigroups, Monatsh. Math. 140 (2003), 103-118] studied various properties of two partial orders on P(X), the semigroup (under composition) consisting of all partial transformations of an arbitrary set X. One partial order was the containment order: namely, ifα,β εP(X) then α⊂ β means x=x for all xdom, the domain of . The other order was the so-called natural order defined by Mitsch [A natural partial order for semigroups, Proc. Amer. Math. Soc. 97(3) (1986), 384-388] for any semigroup. In this paper, we consider these and other orders defined on the symmetric inverse semigroup I(X) and the partial Baer-Levi semigroup PS(q). We show that there are surprising differences between the orders on these semigroups, concerned with their compatibility with respect to composition and the existence of maximal and minimal elements. © 2010 Australian Mathematical Publishing Association Inc.