Baer-Levi semigroups of partial transformations

Let $X$ be an infinite set, and let $q$ be a cardinal number with $\aleph_0\leq q\leq |X|$. The Baer-Levi semigroup on $X$, noted here $BL(q)$, is the set of all injective total transformation $\alpha:X\rightarrow X$ such that $|X\setminus X\alpha| = q$. It is known that $BL(q)$ is a right simple, right cancellative semigroup without idempotents and it is a subsemigroup of the partial Baer-Levi semigroup, denoted by $PS(q)$, consisting of all injective partial transformations $\alpha$ of $X$ such that $|X\setminus X\alpha| = q$. In this talk, we consider some properties of $PS(q)$, concerned with its automorphisms and maximal subsemigroups. We also study the natural partial order defined by Mitsch on the symmetric inverse semigroup $I(X)$ and $PS(q)$.