{"title":"(Λ树公理的(不)依赖性","authors":"Raphael Appenzeller","doi":"10.1515/agms-2023-0106","DOIUrl":null,"url":null,"abstract":"A <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0106_eq_001.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"normal\">Λ</m:mi> </m:math> <jats:tex-math>\\Lambda </jats:tex-math> </jats:alternatives> </jats:inline-formula>-tree is a <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0106_eq_002.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"normal\">Λ</m:mi> </m:math> <jats:tex-math>\\Lambda </jats:tex-math> </jats:alternatives> </jats:inline-formula>-metric space satisfying three axioms (1), (2), and (3). We give a characterization of those ordered abelian groups <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0106_eq_003.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"normal\">Λ</m:mi> </m:math> <jats:tex-math>\\Lambda </jats:tex-math> </jats:alternatives> </jats:inline-formula> for which axioms (1) and (2) imply axiom (3). As a special case, it follows that for the important class of ordered abelian groups <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0106_eq_004.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"normal\">Λ</m:mi> </m:math> <jats:tex-math>\\Lambda </jats:tex-math> </jats:alternatives> </jats:inline-formula> that satisfy <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0106_eq_005.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"normal\">Λ</m:mi> <m:mo>=</m:mo> <m:mn>2</m:mn> <m:mi mathvariant=\"normal\">Λ</m:mi> </m:math> <jats:tex-math>\\Lambda =2\\Lambda </jats:tex-math> </jats:alternatives> </jats:inline-formula>, (3) follows from (1) and (2). For some ordered abelian groups <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0106_eq_006.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"normal\">Λ</m:mi> </m:math> <jats:tex-math>\\Lambda </jats:tex-math> </jats:alternatives> </jats:inline-formula>, we show that axiom (2) is independent of axioms (1) and (3) and ask whether this holds for all ordered abelian groups. Part of this work has been formalized in the proof assistant <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0106_eq_007.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"sans-serif\">Lean</m:mi> </m:math> <jats:tex-math>{\\mathsf{Lean}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"(In)dependence of the axioms of Λ-trees\",\"authors\":\"Raphael Appenzeller\",\"doi\":\"10.1515/agms-2023-0106\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"A <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_agms-2023-0106_eq_001.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi mathvariant=\\\"normal\\\">Λ</m:mi> </m:math> <jats:tex-math>\\\\Lambda </jats:tex-math> </jats:alternatives> </jats:inline-formula>-tree is a <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_agms-2023-0106_eq_002.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi mathvariant=\\\"normal\\\">Λ</m:mi> </m:math> <jats:tex-math>\\\\Lambda </jats:tex-math> </jats:alternatives> </jats:inline-formula>-metric space satisfying three axioms (1), (2), and (3). We give a characterization of those ordered abelian groups <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_agms-2023-0106_eq_003.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi mathvariant=\\\"normal\\\">Λ</m:mi> </m:math> <jats:tex-math>\\\\Lambda </jats:tex-math> </jats:alternatives> </jats:inline-formula> for which axioms (1) and (2) imply axiom (3). As a special case, it follows that for the important class of ordered abelian groups <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_agms-2023-0106_eq_004.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi mathvariant=\\\"normal\\\">Λ</m:mi> </m:math> <jats:tex-math>\\\\Lambda </jats:tex-math> </jats:alternatives> </jats:inline-formula> that satisfy <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_agms-2023-0106_eq_005.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi mathvariant=\\\"normal\\\">Λ</m:mi> <m:mo>=</m:mo> <m:mn>2</m:mn> <m:mi mathvariant=\\\"normal\\\">Λ</m:mi> </m:math> <jats:tex-math>\\\\Lambda =2\\\\Lambda </jats:tex-math> </jats:alternatives> </jats:inline-formula>, (3) follows from (1) and (2). For some ordered abelian groups <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_agms-2023-0106_eq_006.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi mathvariant=\\\"normal\\\">Λ</m:mi> </m:math> <jats:tex-math>\\\\Lambda </jats:tex-math> </jats:alternatives> </jats:inline-formula>, we show that axiom (2) is independent of axioms (1) and (3) and ask whether this holds for all ordered abelian groups. Part of this work has been formalized in the proof assistant <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_agms-2023-0106_eq_007.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi mathvariant=\\\"sans-serif\\\">Lean</m:mi> </m:math> <jats:tex-math>{\\\\mathsf{Lean}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>.\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-04-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1515/agms-2023-0106\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/agms-2023-0106","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
A Λ\Lambda -tree is a Λ\Lambda -metric space satisfying three axioms (1), (2), and (3). We give a characterization of those ordered abelian groups Λ\Lambda for which axioms (1) and (2) imply axiom (3). As a special case, it follows that for the important class of ordered abelian groups Λ\Lambda that satisfy Λ=2Λ\Lambda =2\Lambda , (3) follows from (1) and (2). For some ordered abelian groups Λ\Lambda , we show that axiom (2) is independent of axioms (1) and (3) and ask whether this holds for all ordered abelian groups. Part of this work has been formalized in the proof assistant Lean{\mathsf{Lean}}.