{"title":"Uniformities and a quantale structure on localic groups","authors":"J. Picado, A. Pultr","doi":"10.36045/j.bbms.210823","DOIUrl":null,"url":null,"abstract":"of their size), the idea of shifting neighbourhoods of the group unit via the translation homeomorphisms do not work. Despite this difficulty, the paper proves that there are again induced uniformities (both of Tukey or Weil type) and the two are equivalent. The major tool in the paper is that of a special type of involutive quantale (called G -quantale ), which is borne out of the group structure of the locale. Such quantales are investigated and characterised paving the path to formulating proper-ties of the enriched quantales from which the group structure can be reconstructed. The localic group homomorphisms then turn out to be in one to one correspondence with quantale homomorphisms. Finally, the G -quantales are also ordered semigroups and hence can be used as a set of values for a generalised metric. The paper shows the natural group uniformities are metric uniformities of thus generalised metrics.","PeriodicalId":55309,"journal":{"name":"Bulletin of the Belgian Mathematical Society-Simon Stevin","volume":"91 1","pages":""},"PeriodicalIF":0.4000,"publicationDate":"2022-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bulletin of the Belgian Mathematical Society-Simon Stevin","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.36045/j.bbms.210823","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
of their size), the idea of shifting neighbourhoods of the group unit via the translation homeomorphisms do not work. Despite this difficulty, the paper proves that there are again induced uniformities (both of Tukey or Weil type) and the two are equivalent. The major tool in the paper is that of a special type of involutive quantale (called G -quantale ), which is borne out of the group structure of the locale. Such quantales are investigated and characterised paving the path to formulating proper-ties of the enriched quantales from which the group structure can be reconstructed. The localic group homomorphisms then turn out to be in one to one correspondence with quantale homomorphisms. Finally, the G -quantales are also ordered semigroups and hence can be used as a set of values for a generalised metric. The paper shows the natural group uniformities are metric uniformities of thus generalised metrics.
期刊介绍:
The Bulletin of the Belgian Mathematical Society - Simon Stevin (BBMS) is a peer-reviewed journal devoted to recent developments in all areas in pure and applied mathematics. It is published as one yearly volume, containing five issues.
The main focus lies on high level original research papers. They should aim to a broader mathematical audience in the sense that a well-written introduction is attractive to mathematicians outside the circle of experts in the subject, bringing motivation, background information, history and philosophy. The content has to be substantial enough: short one-small-result papers will not be taken into account in general, unless there are some particular arguments motivating publication, like an original point of view, a new short proof of a famous result etc.
The BBMS also publishes expository papers that bring the state of the art of a current mainstream topic in mathematics. Here it is even more important that at leat a substantial part of the paper is accessible to a broader audience of mathematicians.
The BBMS publishes papers in English, Dutch, French and German. All papers should have an abstract in English.