(In)dependence of the axioms of Λ-trees

Pub Date : 2024-04-01 DOI:10.1515/agms-2023-0106
Raphael Appenzeller
{"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}
引用次数: 0

Abstract

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 = 2 Λ \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}} .
分享
查看原文
(Λ树公理的(不)依赖性
一个Λ \Lambda 树是一个满足三个公理(1)、(2)和(3)的Λ \Lambda 度量空间。我们给出了公理(1)和(2)意味着公理(3)的有序无边群Λ \Lambda的特征。作为一个特例,对于满足Λ = 2 Λ \Lambda =2 \Lambda 的有序边群Λ \Lambda 这一类重要的有序边群,公理(3)是由公理(1)和(2)得出的。对于某些有序无边群Λ \Lambda ,我们证明公理(2)与公理(1)和(3)无关,并询问这是否对所有有序无边群都成立。这项工作的一部分已经在证明助手 Lean {mathsf{Lean} 中正式化了。} .
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信