{"title":"Hyperplanes in matroids and the axiom of choice","authors":"Marianne Morillon","doi":"10.14712/1213-7243.2023.010","DOIUrl":null,"url":null,"abstract":". We show that in set-theory without the axiom of choice ZF , the statement sH : (cid:16)Every proper closed subset of a (cid:28)nitary matroid is the intersection of hyperplanes including it(cid:17) implies AC fin , the axiom of choice for (nonempty) (cid:28)nite sets. We also provide an equivalent of the statement AC fin in terms of (cid:16)graphic(cid:17) matroids. Several open questions stay open in ZF , for example: does sH imply the Axiom of Choice?","PeriodicalId":44396,"journal":{"name":"Commentationes Mathematicae Universitatis Carolinae","volume":" ","pages":""},"PeriodicalIF":0.2000,"publicationDate":"2023-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Commentationes Mathematicae Universitatis Carolinae","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.14712/1213-7243.2023.010","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
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Abstract
. We show that in set-theory without the axiom of choice ZF , the statement sH : (cid:16)Every proper closed subset of a (cid:28)nitary matroid is the intersection of hyperplanes including it(cid:17) implies AC fin , the axiom of choice for (nonempty) (cid:28)nite sets. We also provide an equivalent of the statement AC fin in terms of (cid:16)graphic(cid:17) matroids. Several open questions stay open in ZF , for example: does sH imply the Axiom of Choice?
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