{"title":"群体的粗略选择","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":"{\"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}","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}
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.