几类基于L*上二元聚合算子的蕴涵

L*是区间值模糊与Atanassov意义下的直觉模糊集的基本格。本文首先基于单位区间上的三角模与三角余模,引入L*上两组对偶的二元聚合算子,然后,类似于剩余蕴涵与强蕴涵的构成方法,利用引入的对偶聚合算子生成几类L*上的蕴涵,并对其性质进行讨论。     

正规区间值模糊集上的三角范数

在正规区间值模糊集空间上 ,通过引进区间数的运算及 t-范数算子 ,给出了扩张运算及序的定义 ,讨论了它们的基本性质 .从而 ,获得了这种区间值模糊集关于 t-范数的一些基本结果 .     

A novel approach to interval-valued intuitionistic fuzzy soft set based decision making

The soft set theory, originally proposed by Molodtsov, can be used as a general mathematical tool for dealing with uncertainty. The interval-valued intuitionistic fuzzy soft set is a combination of an interval-valued intuitionistic fuzzy set and a soft set. The aim of this paper is to investigate the decision making based on interval-valued intuitionistic fuzzy soft sets. By means of level soft sets, we develop an adjustable approach to interval-valued intuitionistic fuzzy soft sets based decision making and some numerical examples are provided to illustrate the developed approach. Furthermore, we also define the concept of the weighted interval-valued intuitionistic fuzzy soft set and apply it to decision making.     

格蕴涵代数的区间值$(\in, \in\vee\,q)$-模糊$LI$-理想格

In the present paper, the interval-valued (∈,∈∨q)-fuzzy LI-ideal theory in lattice implication algebras is further studied. Some new properties of interval-valued (∈,∈∨q)-fuzzy LI-ideals are given. Representation theorem of interval-valued (∈,∈∨q)-fuzzy LI-ideal which is generated by an interval-valued fuzzy set is established. It is proved that the set consisting of all interval-valued (∈,∈∨q)-fuzzy LI-ideals in a lattice implication algebra, under the partial order ?, forms a complete distributive lattice.     

The chain and substitution rules of interval-valued intuitionistic fuzzy calculus

The interval-valued intuitionistic fuzzy set proposed by Atanassov is the extension of intuitionistic fuzzy set. It extends the membership degree and non-membership to interval values instead of a single value. So it contains more possible values and maybe more considerate. Among all the researches, the exploration on the calculus of interval-valued intuitionistic fuzzy set is entirely new. Recently, Zhao et al. (Int J Comput Intell Syst 9:36–56, 2016) proposed the concept of interval-valued intuitionistic fuzzy function (IVIFF) and gave a calculation method of derivative and differential of IVIFF. Based on this work, in this paper, firstly, we utilize a new and easier method to express the derivative and differential of IVIFF. Secondly, we propose the chain rules of derivative and the form invariance of differential in the interval-valued intuitionistic fuzzy environment. In addition, some properties of the substation rules for interval-valued intuitionistic fuzzy indefinite integrals and definite integrals are also developed.     

On characterization of generalized interval-valued fuzzy rough sets on two universes of discourse

This paper proposes a general study of (I,T)-interval-valued fuzzy rough sets on two universes of discourse integrating the rough set theory with the interval-valued fuzzy set theory by constructive and axiomatic approaches. Some primary properties of interval-valued fuzzy logical operators and the construction approaches of interval-valued fuzzy T-similarity relations are first introduced. Determined by an interval-valued fuzzy triangular norm and an interval-valued fuzzy implicator, a pair of lower and upper generalized interval-valued fuzzy rough approximation operators with respect to an arbitrary interval-valued fuzzy relation on two universes of discourse is then defined. Properties of I-lower and T-upper interval-valued fuzzy rough approximation operators are examined based on the properties of interval-valued fuzzy logical operators discussed above. Connections between interval-valued fuzzy relations and interval-valued fuzzy rough approximation operators are also established. Finally, an operator-oriented characterization of interval-valued fuzzy rough sets is proposed, that is, interval-valued fuzzy rough approximation operators are characterized by axioms. Different axiom sets of I-lower and T-upper interval-valued fuzzy set-theoretic operators guarantee the existence of different types of interval-valued fuzzy relations which produce the same operators.     

双论域上的区间值直觉模糊粗糙集模型及其应用

基于区间值直觉模糊相容关系,给出了双论域上的区间值直觉模糊粗糙集模型并讨论了其相关性质,为粗糙集的应用提供了新的理论基础与操作手段。最后,通过一个例子阐述了本文提出的区间值直觉模糊粗糙集模型在临床诊断系统中的具体应用。     

TH型区间值模糊正规子群

本文在区间值模糊集空间上，引入了幂等区间范数ＴＨ，在此基础上，定义了ＴＨ型区间值模糊正规子群，并研究了它的一些性质和结构特征，从而拓广了区间值模糊集的理论。     

区间值模糊集和区间值模糊推理

本文在单位区间的一切子区间集Ⅰ[0,1]上引入t-范、t-余范、交(∩)、并(∪)等概念,在此基础上定义了区间值模糊集中的新的交并运算和区间值模糊关系的新的合成运算,得出若干有趣的性质,并将其应用于区间值模糊推理的合成规则上。     

Information granulation and uncertainty measures in interval-valued intuitionistic fuzzy information systems

Information granulation and entropy are main approaches for investigating the uncertainty of information systems, which have been widely employed in many practical domains. In this paper, information granulation and uncertainty measures for interval-valued intuitionistic fuzzy binary granular structures are addressed. First, we propose the representation of interval-valued intuitionistic fuzzy information granules and examine some operations of interval-valued intuitionistic fuzzy granular structures. Second, the interval-valued intuitionistic fuzzy information granularity is introduced to depict the distinguishment ability of an interval-valued intuitionistic fuzzy granular structure (IIFGS), which is a natural extension of fuzzy information granularity. Third, we discuss how to scale the uncertainty of an IIFGS using the extended information entropy and the uncertainty among interval-valued intuitionistic fuzzy granular structures using the expanded mutual information derived from the presented intuitionistic fuzzy information entropy. Fourth, we discovery the relationship between the developed interval-valued intuitionistic fuzzy information entropy and the intuitionistic fuzzy information granularity presented in this paper.     

Exponential distance-based fuzzy clustering for interval-valued data

In several real life and research situations data are collected in the form of intervals, the so called interval-valued data. In this paper a fuzzy clustering method to analyse interval-valued data is presented. In particular, we address the problem of interval-valued data corrupted by outliers and noise. In order to cope with the presence of outliers we propose to employ a robust metric based on the exponential distance in the framework of the Fuzzy C-medoids clustering mode, the Fuzzy C-medoids clustering model for interval-valued data with exponential distance. The exponential distance assigns small weights to outliers and larger weights to those points that are more compact in the data set, thus neutralizing the effect of the presence of anomalous interval-valued data. Simulation results pertaining to the behaviour of the proposed approach as well as two empirical applications are provided in order to illustrate the practical usefulness of the proposed method.     

Natural negation of interval-valued <Emphasis Type="Italic">t</Emphasis>-(co) norms and implications

In this paper, we investigate interval-valued fuzzy negations induced by interval-valued \(t\)-norms, \(t\)-conorms or implications. Some properties of interval-valued fuzzy negations induced by interval-valued sup-morphism \(t\)-norms, inf-morphism \(t\)-conorms or \(R\)-implications are firstly obtained. We also show interval-valued automorphisms acting on the interval-valued fuzzy negations induced by interval-valued \(t\)-norms, \(t\)-conorms or implications. Finally, the relations among the interval-valued fuzzy negations induced by interval-valued \(t\)-norms, \(t\)-conorms or implications are explored.     

D.C.隶属函数模糊集及其应用(I)--D.C.隶属函数模糊集基本概念与性质

本文是D.C.隶属函数模糊集及其应用系列研究的第一部分.建立了D.C.隶属函数模糊集的基本概念.探讨了D.C.隶属函数模糊集的基本性质和D.C.隶属函数模糊集对一些常见的重要t模、余模和伪补的封闭性.并以此建立了丰富的模糊数学应用模型.     

区间值模糊集上的上(下)近似

定义区间值模糊粗糙集的上下近似,并且利用区间值模糊集的截集定义了区间值模糊粗糙度量。并且讨论了它们的性质。     

基于区间数度量的区间值模糊集合的归一化距离、相似度、模糊度和包含度的关系研究

区间值模糊集合的距离、相似度、模糊度和包含度及其关系研究是区间值模糊集合的一个研究热点.考虑到区间值模糊集合所表示信息的丰富性,本文使用区间数而非实数来刻画区间值模糊集合的距离,首先给出基于区间数度量的区间值模糊集合的归一化距离的公理化定义,然后通过五个定理详细研究了基于公理化定义的区间值模糊集合的归一化距离、相似度、模糊度和包含度之间的相互转换关系,最后,给出了若干公式来计算基于区间数度量的区间值模糊集合的相似度、模糊度和包含度.这些结论,一方面丰富了区间值模糊集合的信息测度(距离、相似度、模糊度和包含度)的内容,另一方面也为区间值模糊集合的近似推理、决策分析、模式识别等领域的应用提供了新方法和新理论.     

Operations and measurement of interval-valued fuzzy numbers based on the method of structured element

In this paper we introduced the definition of interval-valued fuzzy numbers based on the definition of interval-valued fuzzy set and the operations of real fuzzy numbers by extension principle, defined the operations of interval-valued fuzzy numbers and their distance, and gave their structured element representations and operations of them by using the fuzzy structured element method. The work of this paper is an expansion of the research of real fuzzy numbers.     

模糊幂格

本文研究了格向其模糊幂集上提升的问题.利用模糊集理论,引入了模糊幂格的概念,获得了模糊幂格及其模糊理想的若干基本性质,推广了格的结果.     

On the continuity points of left-continuous t-norms

Left-continuous t-norms are much more complicated than the continuous ones, and obtaining a complete classification of them seems to be a very hard task. In this paper we investigate some aspects of left-continuous t-norms, with emphasis on their continuity points. In particular, we are interested in left-continuous t-norms which are isomorphic to t-norms which are continuous in the rationals. We characterize such a class, and we prove that it contains the class of all weakly cancellative left-continuous t-norms. S. Jenei was supported by the National Scientific Research Fund Hungary (OTKA F/032782) Mathematics Subject Classification (2000):20M14, 06F05     

The lambda selections of parametric interval-valued fuzzy variables and their numerical characteristics

To model the uncertainty in the secondary possibility distributions, this paper develops a new method for handling interval-valued fuzzy variables with variable lower and upper possibility distributions. For a parametric interval-valued fuzzy variable, we define its lower selection variable, upper selection variable and lambda selection variable. The three selection variables are characterized by variable possibility distributions, and their numerical characteristics like expected values and n-th moments are important indices in practical optimization and decision-making problems. Under this consideration, we establish some useful analytical expressions of the expected values and n-th moments for the lambda selections of parametric interval-valued trapezoidal, normal and Erlang fuzzy variables. Furthermore, we focus on the arithmetic about the sums of common parametric interval-valued fuzzy variables. Finally, we apply the proposed optimization indices to a quantitative finance problem, where the second moment is used to measure the risk of a portfolio.     

Interval-valued level cut sets of fuzzy set

The connections between Zadeh fuzzy set and three-valued fuzzy set are established in this paper. The concepts of interval-valued level cut sets on Zadeh fuzzy set are presented and new decomposition theorems and representation theorems of Zadeh fuzzy set are established based on new cut sets. Firstly, four interval-valued level cut sets on Zadeh fuzzy set are defined as three-valued fuzzy sets and it is shown that the interval-valued level cut sets of Zadeh fuzzy set are generalizations of normal cut sets on Zadeh fuzzy set, and have the same properties as those of normal cut sets of Zadeh fuzzy set. Secondly, the new decomposition theorems are established based on these new cut sets. It is pointed out that each kind of interval-valued level cut sets corresponds to two decomposition theorems. Thus eight decomposition theorems are obtained. Finally, the definitions of three-valued inverse order nested sets and three-valued order nested sets are presented with eight representation theorems based on new nested sets.