Bulletin of the
Korean Mathematical Society BKMS

In 2003, Marques-Smith and Sullivan described the join $\Omega$ of the `natural order' $\leq$ and the `containment order' $\subseteq$ on $P(X)$, the semigroup under composition of all partial transformations of a set $X$. And, in 2004, Pinto and Sullivan described all automorphisms of $PS(q)$, the partial Baer-Levi semigroup consisting of all injective $\alpha\in P(X)$ such that $|X\setminus X\alpha| = q$, where $\aleph_0 \leq q \leq |X|$. In this paper, we describe the group of automorphisms of $R(q)$, the largest regular subsemigroup of $PS(q)$. In 2010, we studied some properties of $\leq $ and $\subseteq$ on $PS(q)$. Here, we characterize the meet and join under those orders for elements of $R(q)$ and $PS(q)$. In addition, since $\leq$ does not equal $\Omega$ on $I(X)$, the symmetric inverse semigroup on $X$, we formulate an algebraic version of $\Omega$ on arbitrary inverse semigroups and discuss some of its properties in an algebraic setting.

Keywords: partial transformation semigroup, Baer-Levi semigroup, inverse semigroup, natural order, containment order, meet and join

MSC numbers: Primary 20M20; Secondary 06A06, 20M18