{"title":"正切配合物和Diamond引理","authors":"Vladimir Dotsenko, Pedro Tamaroff","doi":"10.1142/s1664360723500133","DOIUrl":null,"url":null,"abstract":"The celebrated Diamond Lemma of Bergman gives an effectively verifiable criterion of uniqueness of normal forms for term rewriting in associative algebras. We present a new way to interpret and prove this result from the viewpoint of homotopical algebra. Our main result states that every multiplicative free resolution of an algebra with monomial relations gives rise to its own Diamond Lemma, so that Bergman's condition of resolvable ambiguities becomes the first non-trivial component of the Maurer--Cartan equation in the corresponding tangent complex. The same approach works for many other algebraic structures, emphasizing the relevance of computing multiplicative free resolutions of algebras with monomial relations.","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":"74 1","pages":"0"},"PeriodicalIF":1.1000,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"Tangent complexes and the Diamond Lemma\",\"authors\":\"Vladimir Dotsenko, Pedro Tamaroff\",\"doi\":\"10.1142/s1664360723500133\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The celebrated Diamond Lemma of Bergman gives an effectively verifiable criterion of uniqueness of normal forms for term rewriting in associative algebras. We present a new way to interpret and prove this result from the viewpoint of homotopical algebra. Our main result states that every multiplicative free resolution of an algebra with monomial relations gives rise to its own Diamond Lemma, so that Bergman's condition of resolvable ambiguities becomes the first non-trivial component of the Maurer--Cartan equation in the corresponding tangent complex. The same approach works for many other algebraic structures, emphasizing the relevance of computing multiplicative free resolutions of algebras with monomial relations.\",\"PeriodicalId\":9348,\"journal\":{\"name\":\"Bulletin of Mathematical Sciences\",\"volume\":\"74 1\",\"pages\":\"0\"},\"PeriodicalIF\":1.1000,\"publicationDate\":\"2023-10-13\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Bulletin of Mathematical Sciences\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1142/s1664360723500133\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bulletin of Mathematical Sciences","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1142/s1664360723500133","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
The celebrated Diamond Lemma of Bergman gives an effectively verifiable criterion of uniqueness of normal forms for term rewriting in associative algebras. We present a new way to interpret and prove this result from the viewpoint of homotopical algebra. Our main result states that every multiplicative free resolution of an algebra with monomial relations gives rise to its own Diamond Lemma, so that Bergman's condition of resolvable ambiguities becomes the first non-trivial component of the Maurer--Cartan equation in the corresponding tangent complex. The same approach works for many other algebraic structures, emphasizing the relevance of computing multiplicative free resolutions of algebras with monomial relations.
期刊介绍:
The Bulletin of Mathematical Sciences, a peer-reviewed, open access journal, will publish original research work of highest quality and of broad interest in all branches of mathematical sciences. The Bulletin will publish well-written expository articles (40-50 pages) of exceptional value giving the latest state of the art on a specific topic, and short articles (up to 15 pages) containing significant results of wider interest. Most of the expository articles will be invited.
The Bulletin of Mathematical Sciences is launched by King Abdulaziz University, Jeddah, Saudi Arabia.