On semitopological simple inverse $ω$-semigroups with compact maximal subgroups
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866918060846219264 |
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| author | Gutik, Oleg Maksymyk, Kateryna |
| author_facet | Gutik, Oleg Maksymyk, Kateryna |
| contents | We describe the structure of ($0$-)simple inverse Hausdorff semitopological $ω$-semigroups with compact maximal subgroups. In particular, we show that if $S$ is a simple inverse Hausdorff semitopological $ω$-semigroup with compact maximal subgroups, then $S$ is topologically isomorphic to the Bruck--Reilly extension $\left(\textbf{BR}(T,θ),τ_{\textbf{BR}}^{\oplus}\right)$ of a finite semilattice $T=\left[E;G_α,φ_{α,β}\right]$ of compact groups $G_α$ in the class of topological inverse semigroups, where $τ_{\textbf{BR}}^{\oplus}$ is the sum direct topology on $\textbf{BR}(T,θ)$. Also we prove that every Hausdorff locally compact shift-continuous topology on the simple inverse Hausdorff semitopological $ω$-semigroups with compact maximal subgroups with adjoined zero is either compact or the zero is an isolated point. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_06344 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On semitopological simple inverse $ω$-semigroups with compact maximal subgroups Gutik, Oleg Maksymyk, Kateryna Group Theory General Topology 20M18, 22A15, 54A10, 54D30, 54D40, 54D45, 54H11 We describe the structure of ($0$-)simple inverse Hausdorff semitopological $ω$-semigroups with compact maximal subgroups. In particular, we show that if $S$ is a simple inverse Hausdorff semitopological $ω$-semigroup with compact maximal subgroups, then $S$ is topologically isomorphic to the Bruck--Reilly extension $\left(\textbf{BR}(T,θ),τ_{\textbf{BR}}^{\oplus}\right)$ of a finite semilattice $T=\left[E;G_α,φ_{α,β}\right]$ of compact groups $G_α$ in the class of topological inverse semigroups, where $τ_{\textbf{BR}}^{\oplus}$ is the sum direct topology on $\textbf{BR}(T,θ)$. Also we prove that every Hausdorff locally compact shift-continuous topology on the simple inverse Hausdorff semitopological $ω$-semigroups with compact maximal subgroups with adjoined zero is either compact or the zero is an isolated point. |
| title | On semitopological simple inverse $ω$-semigroups with compact maximal subgroups |
| topic | Group Theory General Topology 20M18, 22A15, 54A10, 54D30, 54D40, 54D45, 54H11 |
| url | https://arxiv.org/abs/2409.06344 |