{"title":"Algebraic representations of (fuzzy) completely distributive lattices","authors":"Xiaowei Wei , Yueli Yue","doi":"10.1016/j.fss.2024.109245","DOIUrl":null,"url":null,"abstract":"<div><div>Choosing completely distributive lattices as lattice-valued environment, this paper introduces the strong <em>L</em>-upper set monad <span><math><mi>S</mi></math></span>, studies Eilenberg-Moore algebras, Kleisli monoids and its applications of <span><math><mi>S</mi></math></span>. Concretely, we give the monad <span><math><mi>S</mi></math></span> by adjoints. Then we characterize fuzzy completely distributive lattices by its Eilenberg-Moore algebras and study Kleisli monoids of the monad. Moreover, we investigate the necessary and sufficient conditions for the establishment of <span><math><mi>S</mi></math></span>. Finally, we construct two monads, study the relationships between these two monads and <span><math><mi>S</mi></math></span>, moreover verify the rationality of the selection of lattice-valued environment by those relationships.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"503 ","pages":"Article 109245"},"PeriodicalIF":3.2000,"publicationDate":"2024-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Fuzzy Sets and Systems","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0165011424003919","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 0
Abstract
Choosing completely distributive lattices as lattice-valued environment, this paper introduces the strong L-upper set monad , studies Eilenberg-Moore algebras, Kleisli monoids and its applications of . Concretely, we give the monad by adjoints. Then we characterize fuzzy completely distributive lattices by its Eilenberg-Moore algebras and study Kleisli monoids of the monad. Moreover, we investigate the necessary and sufficient conditions for the establishment of . Finally, we construct two monads, study the relationships between these two monads and , moreover verify the rationality of the selection of lattice-valued environment by those relationships.
期刊介绍：
Since its launching in 1978, the journal Fuzzy Sets and Systems has been devoted to the international advancement of the theory and application of fuzzy sets and systems. The theory of fuzzy sets now encompasses a well organized corpus of basic notions including (and not restricted to) aggregation operations, a generalized theory of relations, specific measures of information content, a calculus of fuzzy numbers. Fuzzy sets are also the cornerstone of a non-additive uncertainty theory, namely possibility theory, and of a versatile tool for both linguistic and numerical modeling: fuzzy rule-based systems. Numerous works now combine fuzzy concepts with other scientific disciplines as well as modern technologies.
In mathematics fuzzy sets have triggered new research topics in connection with category theory, topology, algebra, analysis. Fuzzy sets are also part of a recent trend in the study of generalized measures and integrals, and are combined with statistical methods. Furthermore, fuzzy sets have strong logical underpinnings in the tradition of many-valued logics.
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