Submonoids of Infinite Symmetric Inverse Monoids

In this thesis we study the subsemigroup structure of the symmetric inverse monoid $I_X$, the inverse semigroup of bijections between subsets of the set $X$, when $X$ is an infinite set. We explore three different approaches to this task. First, we classify the maximal subsemigroups of $I_X$ containing certain subgroups of the symmetric group on $X$. The subgroups in question are the symmetric group itself, the pointwise stabiliser of a finite non-empty subset of $X$, the stabiliser of an ultrafilter on $X$, and the stabiliser of a finite partition of $X$. Next, we study subsemigroups of $I_X$ which are closed in semigroup topologies on $I_X$ introduced by Elliot et al. in 2023. We discover that the closed subsemigroups in these topologies that contain all the idempotents of $I_X$ coincide exactly with semigroups of partial endomorphisms and partial automorphisms of relational structures defined on $X$. Furthermore, we show that if a relational structure $R$ on a countable set $X$ only contains a finite number of relations, then there exists a finite subset $U$ of $I_X$ such that the union of the partial automorphisms of $R$ together with $U$ generates all of $I_X$. Finally, we study the subsemigroup structure of $I_X$ under a preorder introduced by George Bergman and Saharon Shelah in 2006 for the symmetric group. Extending the preorder to $I_X$, if $S_1$ and $S_2$ are subsemigroups of $I_X$, we say that $S_1 \preccurlyeq S_2$ if there exists a finite subset $U$ of $I_X$ such that $S_1$ is contained in the semigroup generated by the union of $S_2$ and $U$. We classify certain types of subsemigroups of $I_X$ according the Bergman-Shelah preorder, and we formulate a conjecture analogous to the main result by Bergman and Shelah.