{"title":"The d-elements of precoherent preidempotent quantales and their applications","authors":"Xianglong Ruan","doi":"10.1093/jigpal/jzae063","DOIUrl":null,"url":null,"abstract":"\n In this paper, we introduce the notion of d-elements on precoherent preidempotent quantale (PIQ), construct Zariski topology on $Max(Q_{d})$ and explore its various properties. Firstly, we give a sufficient condition of a topological space $Max(Q_{d})$ being Hausdorff. Secondly, we prove that if $ P=\\mathfrak{B}(P) $ and $ Q=\\mathfrak{B}(Q) $, then $P$ is isomorphic to $Q$ iff $ Max(P_{d}) $ is homeomorphic to $ Max(Q_{d}) $. Moreover, we prove that $ (P\\otimes Q)_{d} $ is isomorphic to $ P_{d} \\otimes Q_{d} $ iff $ P_{d} \\otimes Q_{d}=(P_{d} \\otimes Q_{d})_{d} $. Finally, we prove that the category $ \\textbf{dPFrm} $ is a reflective subcategory of $\\textbf{PIQuant}.$","PeriodicalId":51114,"journal":{"name":"Logic Journal of the IGPL","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Logic Journal of the IGPL","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1093/jigpal/jzae063","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"LOGIC","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we introduce the notion of d-elements on precoherent preidempotent quantale (PIQ), construct Zariski topology on $Max(Q_{d})$ and explore its various properties. Firstly, we give a sufficient condition of a topological space $Max(Q_{d})$ being Hausdorff. Secondly, we prove that if $ P=\mathfrak{B}(P) $ and $ Q=\mathfrak{B}(Q) $, then $P$ is isomorphic to $Q$ iff $ Max(P_{d}) $ is homeomorphic to $ Max(Q_{d}) $. Moreover, we prove that $ (P\otimes Q)_{d} $ is isomorphic to $ P_{d} \otimes Q_{d} $ iff $ P_{d} \otimes Q_{d}=(P_{d} \otimes Q_{d})_{d} $. Finally, we prove that the category $ \textbf{dPFrm} $ is a reflective subcategory of $\textbf{PIQuant}.$
期刊介绍:
Logic Journal of the IGPL publishes papers in all areas of pure and applied logic, including pure logical systems, proof theory, model theory, recursion theory, type theory, nonclassical logics, nonmonotonic logic, numerical and uncertainty reasoning, logic and AI, foundations of logic programming, logic and computation, logic and language, and logic engineering.
Logic Journal of the IGPL is published under licence from Professor Dov Gabbay as owner of the journal.