Maximal Subsemigroups of Infinite Symmetric Inverse Monoids

Abstract

The symmetric inverse monoid $I_X$ on a set $X$ consists of all bijective functions whose domain and range are subsets of $X$ under the usual composition and inversion of partial functions. For an arbitrary infinite set $X$, we classify all maximal subsemigroups and maximal inverse subsemigroups of $I_X$ which contain the symmetric group Sym($X$) or any of the following subgroups of Sym($X$): the pointwise stabiliser of a finite subset of $X$, the stabiliser of an ultrafilter on $X$, or the stabiliser of a partition of $X$ into finitely many parts of equal cardinality.