{"title":"大卡数阿蒂尼群","authors":"Samuel M. Corson, Saharon Shelah","doi":"arxiv-2408.03201","DOIUrl":null,"url":null,"abstract":"A group is Artinian if there is no infinite strictly descending chain of\nsubgroups. Ol'shanskii has asked whether there are Artinian groups of\narbitrarily large cardinality. We reduce this problem to an analogous question,\nregarding universal algebras, asked by J\\'onsson in the 1960s. We provide\nArtinian groups of cardinality $\\aleph_n$ for each natural number $n$. We also\ngive a consistent strong negative answer to the question of Ol'shanskii (from a\nlarge cardinal assumption) as well as a consistent positive answer. Thus,\nanswers to the questions of Ol'shanskii and J\\'onsson are independent of set\ntheory.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Artinian groups of large cardinality\",\"authors\":\"Samuel M. Corson, Saharon Shelah\",\"doi\":\"arxiv-2408.03201\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"A group is Artinian if there is no infinite strictly descending chain of\\nsubgroups. Ol'shanskii has asked whether there are Artinian groups of\\narbitrarily large cardinality. We reduce this problem to an analogous question,\\nregarding universal algebras, asked by J\\\\'onsson in the 1960s. We provide\\nArtinian groups of cardinality $\\\\aleph_n$ for each natural number $n$. We also\\ngive a consistent strong negative answer to the question of Ol'shanskii (from a\\nlarge cardinal assumption) as well as a consistent positive answer. Thus,\\nanswers to the questions of Ol'shanskii and J\\\\'onsson are independent of set\\ntheory.\",\"PeriodicalId\":501306,\"journal\":{\"name\":\"arXiv - MATH - Logic\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-08-06\",\"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-2408.03201\",\"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-2408.03201","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
A group is Artinian if there is no infinite strictly descending chain of
subgroups. Ol'shanskii has asked whether there are Artinian groups of
arbitrarily large cardinality. We reduce this problem to an analogous question,
regarding universal algebras, asked by J\'onsson in the 1960s. We provide
Artinian groups of cardinality $\aleph_n$ for each natural number $n$. We also
give a consistent strong negative answer to the question of Ol'shanskii (from a
large cardinal assumption) as well as a consistent positive answer. Thus,
answers to the questions of Ol'shanskii and J\'onsson are independent of set
theory.