{"title":"具有广义Möbius函数的非结合关联近环","authors":"John H. Johnson, Max Wakefield","doi":"10.26493/1855-3974.2894.b07","DOIUrl":null,"url":null,"abstract":"There is a convolution product on 3-variable partial flag functions of a locally finite poset that produces a generalized M\\\"obius function. Under the product this generalized M\\\"obius function is a one sided inverse of the zeta function and satisfies many generalizations of classical results. In particular we prove analogues of Phillip Hall's Theorem on the M\\\"obius function as an alternating sum of chain counts, Weisner's theorem, and Rota's Crosscut Theorem. A key ingredient to these results is that this function is an overlapping product of classical M\\\"obius functions. Using this generalized M\\\"obius function we define analogues of the characteristic polynomial and M\\\"obius polynomials for ranked lattices. We compute these polynomials for certain families of matroids and prove that this generalized M\\\"obius polynomial has -1 as root if the matroid is modular. Using results from Ardila and Sanchez we prove that this generalized characteristic polynomial is a matroid valuation.","PeriodicalId":49239,"journal":{"name":"Ars Mathematica Contemporanea","volume":"32 Suppl 3 1","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2022-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"A non-associative incidence near-ring with a generalized Möbius function\",\"authors\":\"John H. Johnson, Max Wakefield\",\"doi\":\"10.26493/1855-3974.2894.b07\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"There is a convolution product on 3-variable partial flag functions of a locally finite poset that produces a generalized M\\\\\\\"obius function. Under the product this generalized M\\\\\\\"obius function is a one sided inverse of the zeta function and satisfies many generalizations of classical results. In particular we prove analogues of Phillip Hall's Theorem on the M\\\\\\\"obius function as an alternating sum of chain counts, Weisner's theorem, and Rota's Crosscut Theorem. A key ingredient to these results is that this function is an overlapping product of classical M\\\\\\\"obius functions. Using this generalized M\\\\\\\"obius function we define analogues of the characteristic polynomial and M\\\\\\\"obius polynomials for ranked lattices. We compute these polynomials for certain families of matroids and prove that this generalized M\\\\\\\"obius polynomial has -1 as root if the matroid is modular. Using results from Ardila and Sanchez we prove that this generalized characteristic polynomial is a matroid valuation.\",\"PeriodicalId\":49239,\"journal\":{\"name\":\"Ars Mathematica Contemporanea\",\"volume\":\"32 Suppl 3 1\",\"pages\":\"\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2022-04-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Ars Mathematica Contemporanea\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.26493/1855-3974.2894.b07\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Ars Mathematica Contemporanea","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.26493/1855-3974.2894.b07","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
A non-associative incidence near-ring with a generalized Möbius function
There is a convolution product on 3-variable partial flag functions of a locally finite poset that produces a generalized M\"obius function. Under the product this generalized M\"obius function is a one sided inverse of the zeta function and satisfies many generalizations of classical results. In particular we prove analogues of Phillip Hall's Theorem on the M\"obius function as an alternating sum of chain counts, Weisner's theorem, and Rota's Crosscut Theorem. A key ingredient to these results is that this function is an overlapping product of classical M\"obius functions. Using this generalized M\"obius function we define analogues of the characteristic polynomial and M\"obius polynomials for ranked lattices. We compute these polynomials for certain families of matroids and prove that this generalized M\"obius polynomial has -1 as root if the matroid is modular. Using results from Ardila and Sanchez we prove that this generalized characteristic polynomial is a matroid valuation.
期刊介绍:
Ars mathematica contemporanea will publish high-quality articles in contemporary mathematics that arise from the discrete and concrete mathematics paradigm. It will favor themes that combine at least two different fields of mathematics. In particular, we welcome papers intersecting discrete mathematics with other branches of mathematics, such as algebra, geometry, topology, theoretical computer science, and combinatorics. The name of the journal was chosen carefully. Symmetry is certainly a theme that is quite welcome to the journal, as it is through symmetry that mathematics comes closest to art.