On Maximal Subsemigroups of Partial Baer-Levi Semigroups

Suppose that X is an infinite set with |X| ≥ q ≥ א0 and I X is the symmetric inverse semigroup defined on X. In 1984, Levi and Wood determined a class of maximal subsemigroups MA using certain subsets A of X of the Baer-Levi semigroup BL q {α ∈ I X : dom α X and |X \ Xα| q}. Later, in 1995, Hotzel showed that there are many other classes of maximal subsemigroups of BL q , but these are far more complicated to describe. It is known that BL q is a subsemigroup of the partial Baer-Levi semigroup PS q {α ∈ I X : |X \ Xα| q}. In this paper, we characterize all maximal subsemigroups of PS q when |X| > q, and we extend MA to obtain maximal subsemigroups of PS q when |X| q.