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ON EQUIVALENCE RELATIONS INDUCED BY LOCALLY COMPACT ABELIAN POLISH GROUPS

Part of: Locally compact abelian groups Abelian groups Set theory

Published online by Cambridge University Press:  07 June 2023

LONGYUN DING Open the ORCID record for LONGYUN DING [Opens in a new window]  and
YANG ZHENG
LONGYUN DING
Affiliation:
SCHOOL OF MATHEMATICAL SCIENCES AND LPMC NANKAI UNIVERSITY TIANJIN 300071, P.R. CHINA E-mail: [email protected] E-mail: [email protected]
YANG ZHENG
Affiliation:
SCHOOL OF MATHEMATICAL SCIENCES AND LPMC NANKAI UNIVERSITY TIANJIN 300071, P.R. CHINA E-mail: [email protected] E-mail: [email protected]
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Abstract

Given a Polish group G, let $E(G)$ be the right coset equivalence relation $G^{\omega }/c(G)$, 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 $P(\omega )/\mbox {Fin}$ can be embedded into Borel equivalence relations between $E({\mathbb {R}}^n)$ and $E({\mathbb {T}}^n)$.

Keywords

Borel reductionlocally compact abelian Polish groupequivalence relation

MSC classification

Primary: 03E15: Descriptive set theory
Secondary: 22B05: General properties and structure of LCA groups 20K45: Topological methods
Type
Article
Information
The Journal of Symbolic Logic , Volume 90 , Issue 1 , March 2025 , pp. 221 - 236
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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