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. |