{"title":"同态$E_0^\\mathbb{N}$的通用二分法","authors":"Assaf Shani","doi":"arxiv-2408.01261","DOIUrl":null,"url":null,"abstract":"We prove the following dichotomy. Given an analytic equivalence relation $E$,\neither ${E_0^{\\mathbb{N}}}\\leq_B{E}$ or else any Borel homomorphism from\n$E_0^{\\mathbb{N}}$ to $E$ is \"very far from a reduction\", specifically, it\nfactors, on a comeager set, through the projection map\n$(2^{\\mathbb{N}})^{\\mathbb{N}}\\to (2^{\\mathbb{N}})^k$ for some\n$k\\in\\mathbb{N}$. As a corollary, we prove that $E_0^{\\mathbb{N}}$ is a prime\nequivalence relation, answering a question on Clemens.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Generic dichotomy for homomorphisms for $E_0^\\\\mathbb{N}$\",\"authors\":\"Assaf Shani\",\"doi\":\"arxiv-2408.01261\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We prove the following dichotomy. Given an analytic equivalence relation $E$,\\neither ${E_0^{\\\\mathbb{N}}}\\\\leq_B{E}$ or else any Borel homomorphism from\\n$E_0^{\\\\mathbb{N}}$ to $E$ is \\\"very far from a reduction\\\", specifically, it\\nfactors, on a comeager set, through the projection map\\n$(2^{\\\\mathbb{N}})^{\\\\mathbb{N}}\\\\to (2^{\\\\mathbb{N}})^k$ for some\\n$k\\\\in\\\\mathbb{N}$. As a corollary, we prove that $E_0^{\\\\mathbb{N}}$ is a prime\\nequivalence relation, answering a question on Clemens.\",\"PeriodicalId\":501306,\"journal\":{\"name\":\"arXiv - MATH - Logic\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-08-02\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Logic\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2408.01261\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Logic","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.01261","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Generic dichotomy for homomorphisms for $E_0^\mathbb{N}$
We prove the following dichotomy. Given an analytic equivalence relation $E$,
either ${E_0^{\mathbb{N}}}\leq_B{E}$ or else any Borel homomorphism from
$E_0^{\mathbb{N}}$ to $E$ is "very far from a reduction", specifically, it
factors, on a comeager set, through the projection map
$(2^{\mathbb{N}})^{\mathbb{N}}\to (2^{\mathbb{N}})^k$ for some
$k\in\mathbb{N}$. As a corollary, we prove that $E_0^{\mathbb{N}}$ is a prime
equivalence relation, answering a question on Clemens.