Spectral Duality for Finitely Generated Nilpotent Minimum Algebras, with Applications

We establish a categorical duality for the finitely generatedLindenbaum-Tarski algebras of propositional nilpotent minimumlogic. The latter's conjunction is semantically interpretedby a left-continuous (but not continuous) triangular norm; implicationis obtained through residuation. Our duality allows one to transferto nilpotent minimum logic several known results about inutitionisticlogic with the prelinearity axiom (also called Gödel-Dummettlogic), mutatis mutandis. We give several such applications.     

Varieties generated by T-norms

This is a survey of some work done recently on varieties generated by algebras arising in fuzzy set theory and logic. Details and proofs appear elsewhere. The hope is to impart some of the flavor of this particular topic, and to point out some areas that merit further study.     

Iséki algebras. Connection with BL algebras

We introduce Iséki algebras as a special class of BCK algebras and we prove they are categorically equivalent with BL algebras. Other connections are established.I thank very much Professor Antonio Di Nola for his kind suggestions that helped me to improve the final form of the paper.     

Residual operators of uninorms

 Uninorms are an important generalization of t-norms and t-conorms, having a neutral element lying anywhere in the unit interval. A uninorm shows a typical block structure and is built from a t-norm, a t-conorm and a mean operator. Two important classes of uninorms are characterized, corresponding to the use of the minimum operator (the class U _min) and maximum operator (the class U _max) as mean operator. The characterization of representable uninorms, i.e. uninorms with an additive generator, and of left-continuous and right-continuous idempotent uninorms is recalled. Two residual operators are associated with a uninorm and it is characterized when they yield an implicator and coimplicator. The block structure of the residual implicator of members of the class U _min and of the residual coimplicator of members of the class U _max is investigated. Explicit expressions for the residual implicator and residual coimplicator of representable uninorms and of certain left-continuous or right-continuous idempotent uninorms are given. Additional properties such as contrapositivity are discussed.     

Multicriteria decision-making using Archimedean aggregation operators in Pythagorean hesitant fuzzy environment

In multicriteria decision-making (MCDM), the existing aggregation operators are mostly based on algebraic t-conorm and t-norm. But, Archimedean t-conorms and t-norms are the generalized forms of t-conorms and t-norms which include algebraic, Einstein, Hamacher, Frank, and other types of t-conorms and t-norms. From that view point, in this paper the concepts of Archimedean t-conorm and t-norm are introduced to aggregate Pythagorean hesitant fuzzy information. Some new operational laws for Pythagorean hesitant fuzzy numbers based on Archimedean t-conorm and t-norm have been proposed. Using those operational laws, Archimedean t-conorm and t-norm-based Pythagorean hesitant fuzzy weighted averaging operator and weighted geometric operator are developed. Some of their desirable properties have also been investigated. Afterwards, these operators are applied to solve MCDM problems in Pythagorean hesitant fuzzy environment. The developed Archimedean aggregation operators are also applicable in Pythagorean fuzzy contexts also. To demonstrate the validity, practicality, and effectiveness of the proposed method, a practical problem is considered, solved, and compared with other existing method.     

基于知识划分相似性的属性约简

信息系统的属性约简是粗糙集理论的重要内容之一。除正区域、差别矩阵、信息熵之外,运用模糊T的性质提出了一种基于t-范数的划分,基于知识的划分,给出了相似性的概念,提出了若干相似性的性质,并将该相似性的度量运用到属性约简中,给出了一个新的属性约简算法,从而对属性约简进行改进。通过一个数据模型的验证,新的算法同样可以有效地滤除冗余属性,保留关键属性,充分说明了该方法的可行性。     

基于爱因斯坦t-模的模糊联想记忆的学习算法

为神经网络提供有效学习算法是神经网络研究的关键问题。文章利用t-模的伴随蕴涵算子,为基于Max和Tes合成的模糊联想记忆网络Max-TesFAM提供了一种新的学习算法,此处Tes是由爱因斯坦提出的一种t-模算子。从理论上严格证明了,只要Max-TesFAM能完整可靠地存储所给的多个模式对,则该新的学习算法一定能找到使得网络能完整可靠存储这些模式对的所有连接权矩阵的最大者。最后,用实验说明了所提出的学习算法的有效性。     

Cubic fuzzy Einstein aggregation operators and its application to decision-making

In this paper, we define some Einstein operations on cubic fuzzy set (CFS) and develop three arithmetic averaging operators, which are cubic fuzzy Einstein weighted averaging (CFEWA) operator, cubic fuzzy Einstein ordered weighted averaging (CFEOWA) operator and cubic fuzzy Einstein hybrid weighted averaging (CFEHWA) operator, for aggregating cubic fuzzy data. The CFEHWA operator generalises both the CFEWA and CFEOWA operators. Furthermore, we develop various properties of these operators and derive the relationship between the proposed operators and the exiting aggregation operators. We apply CFEHWA operator to multiple attribute decision-making with cubic fuzzy data. Finally, a numerical example is constructed to demonstrate the established approach.     

Alternative Proof of Standard Completeness Theorem for MTL

As it was shown in Jenei and Montagna (Studia Logica 70(2): 183–192, 2002), Monoidal t-norm based logic (MTL) is a logic of left-continuous t-norms. In other words, this means that MTL enjoys the standard completeness theorem. In this paper we present a different proof of this theorem. In fact, we prove even more since we show that MTL is complete w.r.t. the class of standard MTL-algebras with finite congruence lattice or equivalently with finitely many Archimedean classes. We also show the connection between the congruence lattice of an MTL-chain and its Archimedean classes.The author was supported by the Program “Information Society” under project 1ET100300517.     

A characterization of uninorms on bounded lattices by means of triangular norms and triangular conorms

ABSTRACT

Uninorms are generalizations of triangular norms and triangular conorms leaving the freedom for the neutral element to be an arbitrary element from a bounded lattice. In this paper, we study uninorms on bounded lattices and investigate their main characteristics. We also introduce the new construction methods for uninorms on bounded lattices with a neutral element based on the fact that the presence of triangular norms and triangular conorms. Furthermore, we assess and exemplify the differences between our constructions and the present approaches.