PARTIAL ORDERS ON PARTIAL BAER–LEVI SEMIGROUPS
Boorapa Singha, Jintana Sanwong, Robert Patrick Sullivan · Bulletin of the Australian Mathematical Society · 2010
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, if α , β ∈ P ( X ) then α ⊆ β means xα = xβ for all x ∈dom α , the domain of α . 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.