{"title":"拟度量空间的局部Yoneda补全","authors":"Jing Lu, Bin Zhao","doi":"10.1017/S0960129523000105","DOIUrl":null,"url":null,"abstract":"Abstract In this paper, we study quasi-metric spaces using domain theory. Given a quasi-metric space (X,d), we use \n$({\\bf B}(X,d),\\leq^{d^{+}}\\!)$\n to denote the poset of formal balls of the associated quasi-metric space (X,d). We introduce the notion of local Yoneda-complete quasi-metric spaces in terms of domain-theoretic properties of \n$({\\bf B}(X,d),\\leq^{d^{+}}\\!)$\n . The manner in which this definition is obtained is inspired by Romaguera–Valero theorem and Kostanek–Waszkiewicz theorem. Furthermore, we obtain characterizations of local Yoneda-complete quasi-metric spaces via local nets in quasi-metric spaces. More precisely, we prove that a quasi-metric space is local Yoneda-complete if and only if every local net has a d-limit. Finally, we prove that every quasi-metric space has a local Yoneda completion.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"33 1","pages":"33 - 45"},"PeriodicalIF":0.4000,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Local Yoneda completions of quasi-metric spaces\",\"authors\":\"Jing Lu, Bin Zhao\",\"doi\":\"10.1017/S0960129523000105\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract In this paper, we study quasi-metric spaces using domain theory. Given a quasi-metric space (X,d), we use \\n$({\\\\bf B}(X,d),\\\\leq^{d^{+}}\\\\!)$\\n to denote the poset of formal balls of the associated quasi-metric space (X,d). We introduce the notion of local Yoneda-complete quasi-metric spaces in terms of domain-theoretic properties of \\n$({\\\\bf B}(X,d),\\\\leq^{d^{+}}\\\\!)$\\n . The manner in which this definition is obtained is inspired by Romaguera–Valero theorem and Kostanek–Waszkiewicz theorem. Furthermore, we obtain characterizations of local Yoneda-complete quasi-metric spaces via local nets in quasi-metric spaces. More precisely, we prove that a quasi-metric space is local Yoneda-complete if and only if every local net has a d-limit. Finally, we prove that every quasi-metric space has a local Yoneda completion.\",\"PeriodicalId\":49855,\"journal\":{\"name\":\"Mathematical Structures in Computer Science\",\"volume\":\"33 1\",\"pages\":\"33 - 45\"},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2023-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematical Structures in Computer Science\",\"FirstCategoryId\":\"94\",\"ListUrlMain\":\"https://doi.org/10.1017/S0960129523000105\",\"RegionNum\":4,\"RegionCategory\":\"计算机科学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"COMPUTER SCIENCE, THEORY & METHODS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Structures in Computer Science","FirstCategoryId":"94","ListUrlMain":"https://doi.org/10.1017/S0960129523000105","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
Abstract In this paper, we study quasi-metric spaces using domain theory. Given a quasi-metric space (X,d), we use
$({\bf B}(X,d),\leq^{d^{+}}\!)$
to denote the poset of formal balls of the associated quasi-metric space (X,d). We introduce the notion of local Yoneda-complete quasi-metric spaces in terms of domain-theoretic properties of
$({\bf B}(X,d),\leq^{d^{+}}\!)$
. The manner in which this definition is obtained is inspired by Romaguera–Valero theorem and Kostanek–Waszkiewicz theorem. Furthermore, we obtain characterizations of local Yoneda-complete quasi-metric spaces via local nets in quasi-metric spaces. More precisely, we prove that a quasi-metric space is local Yoneda-complete if and only if every local net has a d-limit. Finally, we prove that every quasi-metric space has a local Yoneda completion.
期刊介绍:
Mathematical Structures in Computer Science is a journal of theoretical computer science which focuses on the application of ideas from the structural side of mathematics and mathematical logic to computer science. The journal aims to bridge the gap between theoretical contributions and software design, publishing original papers of a high standard and broad surveys with original perspectives in all areas of computing, provided that ideas or results from logic, algebra, geometry, category theory or other areas of logic and mathematics form a basis for the work. The journal welcomes applications to computing based on the use of specific mathematical structures (e.g. topological and order-theoretic structures) as well as on proof-theoretic notions or results.