Abstract. Let X be an infinite set and suppose that aleph₀≤ 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\X α| = q. They also determined its subsemigroup S(q,r) = {α ∈ PS(q) : |X\dom α| ≤ r}, where aleph₀ ≤ r ≤ |X|. Recently, Singha and Sanwong showed that, when q< |X|, almos 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.

Received: November 15, 2012

AMS Subject Classification: 20M20

Key Words and Phrases: injective partial transformation, partial Baer-Levi semigroup, gap, defect

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