Topological embeddings into transformation monoids
| Autor: | Bardyla, S., Elliott, L., Mitchell, J. D., Peresse, Y. |
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| Rok vydání: | 2023 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | In this paper we consider the questions of which topological semigroups embed topologically into the full transformation monoid $\mathbb{N} ^ \mathbb{N}$ or the symmetric inverse monoid $I_{\mathbb{N}}$ with their respective canonical Polish semigroup topologies. We characterise those topological semigroups that embed topologically into $\mathbb{N} ^ \mathbb{N}$ and satisfy any of the following properties: commutative; compact; groups; countable Polish Clifford; arbitrary Clifford semigroups with certain properties on their semilattice of idempotents. We prove analogous characterisation of these results for topological inverse semigroups and $I_{\mathbb{N}}$. We construct several examples of countable Polish topological semigroups that do not embed into $\mathbb{N} ^ \mathbb{N}$, which answer, in the negative, a recent open problem of Elliott et al. Additionally, we obtain two sufficient conditions for a topological Clifford semigroup to be metrizable, and prove that inversion is automatically continuous in every Clifford subsemigroup of $\mathbb{N}^\mathbb{N}$. The former complements recent works of Banakh et al. Comment: 19 pages |
| Databáze: | arXiv |
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