In 2003, Marques-Smith and Sullivan described the join

of the 'natural order'

and the 'containment order'

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

such that

, where

. 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

and

on PS(q). Here, we characterize the meet and join under those orders for elements of R(q) and PS(q). In addition, since

does not equal

on I(X), the symmetric inverse semigroup on X, we formulate an algebraic version of

on arbitrary inverse semigroups and discuss some of its properties in an algebraic setting.