Resumo:  In 2003, Marques-Smith and Sullivan studied various properties of P(X), the semigroup of all partial transformations of a set X, under two natural partial orders, namely the Mitsch order (1986) and the containment order (Lyapin 1953). In particular, they characterised the meet and join of these orders and determined which elements of P(X) are left (or right) compatible with respect to each order.

There are two interesting subsemigroups of P(X): namely, I(X), the semigroup of all injective (one-to-one) elements of P(X), first studied by Vagner and Preston in the early 1950s; and, when X is infinite, a subsemigroup PS(q) of I(X) which was investigated by Pinto and Sullivan in 2004.

Recently, Singha, Sanwong and Sullivan (2010 and 2011) considered I(X) and PS(q) under the same natural partial orders and answered questions about their meet and join on the respective semigroups, the existence of compatible elements, and so on. In so doing, they discovered a new partial order on an arbitrary inverse semigroup whose significance is still unknown. In this talk, we will discuss some of these ideas.