Injective partial transformations with infinite defects

Abstract

In 2003, Marques-Smith and Sullivan described the join Ω 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 α {small element of} P(X) such that {pipe}X \ Xα{pipe} = q, where א 0 ≤ q ≤ {pipe}X{pipe}. 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 Ω on I(X), the symmetric inverse semigroup on X, we formulate an algebraic version of Ω on arbitrary inverse semigroups and discuss some of its properties in an algebraic setting. © 2012 The Korean Mathematical Society.