Submonoids of Infinite Symmetric Inverse Monoids
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2025
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| author | Hampenberg, Martin |
| author_facet | Hampenberg, Martin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_04200 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Submonoids of Infinite Symmetric Inverse Monoids Hampenberg, Martin Rings and Algebras Combinatorics General Topology Group Theory 20M20 (Primary) 20M18, 05C05, 08A05, 22A15 (Secondary) 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. |
| title | Submonoids of Infinite Symmetric Inverse Monoids |
| topic | Rings and Algebras Combinatorics General Topology Group Theory 20M20 (Primary) 20M18, 05C05, 08A05, 22A15 (Secondary) |
| url | https://arxiv.org/abs/2509.04200 |