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

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

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