{"title":"Semiaffine stable planes","authors":"Rainer Löwen, Markus J. Stroppel","doi":"10.1007/s13366-023-00720-z","DOIUrl":null,"url":null,"abstract":"Abstract A locally compact stable plane of positive topological dimension will be called semiaffine if for every line L and every point p not in L there is at most one line passing through p and disjoint from L . We show that then the plane is either an affine or projective plane or a punctured projective plane (i.e., a projective plane with one point deleted). We also compare this with the situation in general linear spaces (without topology), where P. Dembowski showed that the analogue of our main result is true for finite spaces but fails in general.","PeriodicalId":44678,"journal":{"name":"Beitrage zur Algebra und Geometrie-Contributions to Algebra and Geometry","volume":"109 3","pages":"0"},"PeriodicalIF":0.4000,"publicationDate":"2023-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Beitrage zur Algebra und Geometrie-Contributions to Algebra and Geometry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s13366-023-00720-z","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Abstract A locally compact stable plane of positive topological dimension will be called semiaffine if for every line L and every point p not in L there is at most one line passing through p and disjoint from L . We show that then the plane is either an affine or projective plane or a punctured projective plane (i.e., a projective plane with one point deleted). We also compare this with the situation in general linear spaces (without topology), where P. Dembowski showed that the analogue of our main result is true for finite spaces but fails in general.
期刊介绍:
The Journal "Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry" was founded in 1971 on the occasion of the 65th birthday of O.-H. Keller. It publishes research articles in the areas of algebra, geometry, algebraic geometry and related fields, preferably in English language.
The back issues of the journal are available at the European Digital Mathematics Library (EuDML) at:
https://eudml.org/journal/10170 (Vols. 1-33, 1971-1992)
https://eudml.org/journal/10084 (Vols. 34-51, 1993-2010)