{"title":"Coarse selectors of groups","authors":"Igor Protasov","doi":"10.12958/adm2127","DOIUrl":null,"url":null,"abstract":"For a group G, FG denotes the set of all non-empty finite subsets of G. We extend the finitary coarse structure of G from G×G to FG×FG and say that a macro-uniform mapping f:FG→FG (resp. f:[G]2→G) is a finitary selector (resp. 2-selector) of G if f(A)∈A for each A ∈ FG (resp. A∈[G]2). Weprove that a group G admits a finitary selector if and only if G admits a 2-selector and if and only if G is a finite extension of an infinite cyclic subgroup or G is countable and locally finite. We use this result to characterize groups admitting linear orders compatible with finitary coarse structures.","PeriodicalId":364397,"journal":{"name":"Algebra and Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Algebra and Discrete Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.12958/adm2127","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
For a group G, FG denotes the set of all non-empty finite subsets of G. We extend the finitary coarse structure of G from G×G to FG×FG and say that a macro-uniform mapping f:FG→FG (resp. f:[G]2→G) is a finitary selector (resp. 2-selector) of G if f(A)∈A for each A ∈ FG (resp. A∈[G]2). Weprove that a group G admits a finitary selector if and only if G admits a 2-selector and if and only if G is a finite extension of an infinite cyclic subgroup or G is countable and locally finite. We use this result to characterize groups admitting linear orders compatible with finitary coarse structures.