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ON EQUIVALENCE RELATIONS INDUCED BY LOCALLY COMPACT ABELIAN POLISH GROUPS
Published online by Cambridge University Press: 07 June 2023
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 $E({\mathbb {T}}^n)$ .
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- © The Author(s), 2023. Published by Cambridge University Press on behalf of The Association for Symbolic Logic
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