On equivalence relations induced by locally compact abelian Polish groups
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866908318770921472 |
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| author | Ding, Longyun Zheng, Yang |
| author_facet | Ding, Longyun Zheng, Yang |
| contents | Given a Polish group $G$, let $E(G)$ be the right coset equivalence relation $G^ω/c(G)$, where $c(G)$ is the group of all convergent sequences in $G$. The connected component of the identity of a Polish group $G$ is denoted by $G_0$. Let $G,H$ be locally compact abelian Polish groups. If $E(G)\leq_B E(H)$, then there is a continuous homomorphism $S:G_0\rightarrow H_0$ such that $\ker(S)$ is non-archimedean. The converse is also true when $G$ is connected and compact. For $n\in{\mathbb N}^+$, the partially ordered set $P(ω)/\mbox{Fin}$ can be embedded into Borel equivalence relations between $E({\mathbb R}^n)$ and $E({\mathbb T}^n)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_12167 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On equivalence relations induced by locally compact abelian Polish groups Ding, Longyun Zheng, Yang Logic 03E15 Given a Polish group $G$, let $E(G)$ be the right coset equivalence relation $G^ω/c(G)$, where $c(G)$ is the group of all convergent sequences in $G$. The connected component of the identity of a Polish group $G$ is denoted by $G_0$. Let $G,H$ be locally compact abelian Polish groups. If $E(G)\leq_B E(H)$, then there is a continuous homomorphism $S:G_0\rightarrow H_0$ such that $\ker(S)$ is non-archimedean. The converse is also true when $G$ is connected and compact. For $n\in{\mathbb N}^+$, the partially ordered set $P(ω)/\mbox{Fin}$ can be embedded into Borel equivalence relations between $E({\mathbb R}^n)$ and $E({\mathbb T}^n)$. |
| title | On equivalence relations induced by locally compact abelian Polish groups |
| topic | Logic 03E15 |
| url | https://arxiv.org/abs/2209.12167 |