Reducibility and nonreducibility between <IMG ALIGN=BOTTOM HEIGHT=14 WIDTH=15> equivalence relations期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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Reducibility and nonreducibility between equivalence relations

Authors:	Randall Dougherty Greg Hjorth

Institution:	Department of Mathematics, Ohio State University, Columbus, Ohio 43210 ; Department of Mathematics, University of California, Los Angeles, California 90095-1555

Abstract:	We show that, for $1 \le p < q < \infty$ , the relation of $\ell ^{p}$ -equivalence between infinite sequences of real numbers is Borel reducible to the relation of $\ell ^{q}$ -equivalence (i.e., the Borel cardinality of the quotient ${\mathbb R}^{{\mathbb N}}/\ell ^{p}$ is no larger than that of ${\mathbb R}^{{\mathbb N}}/\ell ^{q}$ ), but not vice versa. The Borel reduction is constructed using variants of the triadic Koch snowflake curve; the nonreducibility in the other direction is proved by taking a putative Borel reduction, refining it to a reduction map that is not only continuous but `modular,' and using this nicer map to derive a contradiction.

Keywords:	Borel equivalence relations reducibility Borel cardinality

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