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  1.  2
    On Equivalence Relations Induced by Locally Compact Abelian Polish Groups.Longyun Ding & Yang Zheng - forthcoming - Journal of Symbolic Logic:1-16.
    Given a Polish groupG, let$E(G)$be the right coset equivalence relation$G^{\omega }/c(G)$, where$c(G)$is the group of all convergent sequences inG. The connected component of the identity of a Polish groupGis 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 whenGis 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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  2.  41
    Borel reducibility and Hölder(α) embeddability between Banach spaces.Longyun Ding - 2012 - Journal of Symbolic Logic 77 (1):224-244.
    We investigate Borel reducibility between equivalence relations $E(X;p)=X^{\mathbb{N}}/\ell_{p}(X)'s$ where X is a separable Banach space. We show that this reducibility is related to the so called Hölder(α) embeddability between Banach spaces. By using the notions of type and cotype of Banach spaces, we present many results on reducibility and unreducibility between E(L r ; p)'s and E(c 0 ; p)'s for r, p ∈ [1, +∞). We also answer a problem presented by Kanovei in the affirmative by showing that $\mathrm{C}\left({\mathrm{\mathbb{R}}}^{+}\right)/{\mathrm{C}}_{0}\left({\mathrm{\mathbb{R}}}^{+}\ri ght)$ (...)
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  3.  42
    Borel reducibility and finitely Hölder (α) embeddability.Longyun Ding - 2011 - Annals of Pure and Applied Logic 162 (12):970-980.
    Let , be a sequence of pseudo-metric spaces, and let p≥1. For , let . For Borel reducibility between equivalence relations , we show it is closely related to finitely Hölder embeddability between pseudo-metric spaces.
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  4.  4
    A Hierarchy on Non-Archimedean Polish Groups Admitting a Compatible Complete Left-Invariant Metric.Longyun Ding & Xu Wang - forthcoming - Journal of Symbolic Logic:1-19.
    In this article, we introduce a hierarchy on the class of non-archimedean Polish groups that admit a compatible complete left-invariant metric. We denote this hierarchy by $\alpha $ -CLI and L- $\alpha $ -CLI where $\alpha $ is a countable ordinal. We establish three results: (1) G is $0$ -CLI iff $G=\{1_G\}$ ; (2) G is $1$ -CLI iff G admits a compatible complete two-sided invariant metric; and (3) G is L- $\alpha $ -CLI iff G is locally $\alpha $ (...)
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  5.  6
    Characterization of ‐like and c 0 ‐like equivalence relations.Longyun Ding - 2014 - Mathematical Logic Quarterly 60 (4-5):273-279.
    In this paper, notions of ‐like and c0‐like equivalence relations are introduced. We characterize the positions of ‐like and c0‐like equivalence relations in the Borel reducibility hierarchy by comparing them with equivalence relations and.
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  6.  12
    Decomposing functions of baire class on polish spaces.Longyun Ding, Takayuki Kihara, Brian Semmes & Jiafei Zhao - 2020 - Journal of Symbolic Logic 85 (3):960-971.
    We prove the Decomposability Conjecture for functions of Baire class $2$ from a Polish space to a separable metrizable space. This partially answers an important open problem in descriptive set theory.
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  7.  12
    On equivalence relations generated by Cauchy sequences in countable metric spaces.Longyun Ding & Kai Gu - 2020 - Annals of Pure and Applied Logic 171 (10):102854.
  8.  24
    Reductions on equivalence relations generated by universal sets.Longyun Ding & Ping Yu - 2019 - Mathematical Logic Quarterly 65 (1):8-13.
    Let X, Y be Polish spaces,,. We say A is universal for Γ provided that each x‐section of A is in Γ and each element of Γ occurs as an x‐section of A. An equivalence relation generated by a set is denoted by, where. The following results are shown: If A is a set universal for all nonempty closed subsets of Y, then is a equivalence relation and. If A is a set universal for all countable subsets of Y, then (...)
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  9.  31
    Diagonal Actions and Borel Equivalence Relations.Longyun Ding & Su Gao - 2006 - Journal of Symbolic Logic 71 (4):1081 - 1096.
    We investigate diagonal actions of Polish groups and the related intersection operator on closed subgroups of the acting group. The Borelness of the diagonal orbit equivalence relation is characterized and is shown to be connected with the Borelness of the intersection operator. We also consider relatively tame Polish groups and give a characterization of them in the class of countable products of countable abelian groups. Finally an example of a logic action is considered and its complexity in the Borel reducbility (...)
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