SEMIGROUP OF INJECTIVE PARTIAL TRANSFORMATIONS WITH BOUNDED GAP AND FIXED DEFECT

Abstract

Let X be an infinite set and suppose that ℵ0 ≤ q ≤ |X|. In 2004, Pinto and Sullivan considered algebraic properties of PS(q), the partial Baer-Levi semigroup consisting of all injective partial transformations α of X such that |X \Xα| = q. They also determined its subsemigroup S(q,r) = {α ∈ PS(q) : |X domα| ≤ r} where ℵ0 ≤ r ≤ |X|. Recently, Singha and Sanwong showed that, when q < |X|, almost every maximal subsemigroup of PS(q) is induced by a maximal subsemigroup of S(q,r). Here, we use their work to describe some algebraic properties of S(q,r) including its Green's relation and its ideal structure.