{"title":"Ordinal sum combinations of continuous t-norms and their related ordered algebraic structures","authors":"Hongjun Zhou","doi":"10.1016/j.fss.2025.109521","DOIUrl":null,"url":null,"abstract":"<div><div>Recently, Vemuri et al. (2023) <span><span>[15]</span></span> proposed two operators for constructing new continuous t-norms with the intent of integrating the ordinal sum structures of given ones. Unfortunately, these two operators on non-trivial ordinal sum t-norms are not well-defined so that almost all the results in that part as well as some ones of the following papers are incorrect. The purpose of this paper is to revisit such topic, to redefine a pair of combination operations and to study their related ordered algebraic structures. It is proved that our redefined operations both satisfy the laws of commutativity, associativity, idempotency, and <em>finer</em> absorption, and take the minimum t-norm as the absorption and neutral elements, respectively. Since these operations will finerize in general the subintervals of the underlying t-norms, it is impossible to make the set of all continuous t-norms a lattice under these two operations, and then we turn to find the sufficient and necessary conditions for these operations to be lattice operations and get finally many distributive sublattices of continuous t-norms. Among four possible pairs of ordinal sum combination operations on continuous t-norms, the pair defined in the text matches every well and induces the richest distributive sublattice structures. Lastly, our redefined operations are applied to the contexts of <span><math><mo>(</mo><mi>S</mi><mo>,</mo><mi>N</mi><mo>)</mo></math></span>-implications and residual implications with continuous t-conorms and t-norms, respectively.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"519 ","pages":"Article 109521"},"PeriodicalIF":3.2000,"publicationDate":"2025-07-07","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/S016501142500260X","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
Recently, Vemuri et al. (2023) [15] proposed two operators for constructing new continuous t-norms with the intent of integrating the ordinal sum structures of given ones. Unfortunately, these two operators on non-trivial ordinal sum t-norms are not well-defined so that almost all the results in that part as well as some ones of the following papers are incorrect. The purpose of this paper is to revisit such topic, to redefine a pair of combination operations and to study their related ordered algebraic structures. It is proved that our redefined operations both satisfy the laws of commutativity, associativity, idempotency, and finer absorption, and take the minimum t-norm as the absorption and neutral elements, respectively. Since these operations will finerize in general the subintervals of the underlying t-norms, it is impossible to make the set of all continuous t-norms a lattice under these two operations, and then we turn to find the sufficient and necessary conditions for these operations to be lattice operations and get finally many distributive sublattices of continuous t-norms. Among four possible pairs of ordinal sum combination operations on continuous t-norms, the pair defined in the text matches every well and induces the richest distributive sublattice structures. Lastly, our redefined operations are applied to the contexts of -implications and residual implications with continuous t-conorms and t-norms, respectively.
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.