广义剩余蕴涵北大核心

本文基于一般的聚合算子,提出一类新的模糊蕴涵,称之为广义剩余蕴涵.广义剩余蕴涵是模糊逻辑中十分重要的剩余蕴涵的自然推广.我们进一步讨论这类蕴涵的基本性质,并且探究广义剩余蕴涵与由模糊合取生成的蕴涵之间的关系,并证明这两类蕴涵为不同的蕴涵.这些结果在模糊逻辑与模糊决策之间建立了新的联系.     

模糊蕴涵格理论

模糊蕴涵代数,在文献中简称为FI代数,最初由吴望名先生于1990年提出,至今已经有许多研究成果.文中综述有关FI代数的概念,性质等主要研究工作,同时给出这类代数的一些新的性质.重点强调构成格结构的FI代数,称之为模糊蕴涵格,简称为FI格.这类代数结构与模糊逻辑中几个重要的代数系统具有紧密的联系,文中将揭示这些联系,一些重要的模糊逻辑代数系统都是FI格类的子类.另外,所有正则FI格构成代数簇,即等式代数类.这个代数簇将在模糊逻辑与近似推理中发挥重要的作用.     

剩余格蕴涵代数中多重模糊结合滤子的研究

在剩余格蕴涵代数中,首先提出了多重模糊结合滤子的概念,研究了其性质.然后,讨论了多重模糊结合滤子与模糊滤子之间的关系;接着,讨论了多重模糊结合滤子与多重模糊关联滤子之间的关系;最后,得到了在格H蕴涵代数中,不同的多重模糊结合滤子之间等价的结论.     

基于S-蕴涵的模糊等价

对由S-蕴涵所定义的模糊等价的性质进行了详细的研究.我们首先以文献中基于R-蕴涵的模糊等价的相关结果为基础,在涉及的t-余模S无任何限制条件下给出了模糊等价一般性质,其次详细利用彭等提出的对De Morgan三元组所给出的限制性条件,给出了基于S-蕴涵的模糊等价相关性质的进一步讨论.所得的结果丰富了模糊等价的理论研究.     

格蕴涵代数的区间值模糊子代数

将区间值模糊集的概念应用于格蕴涵代数,引入区间值模糊格蕴涵子代数的概念并研究它们的性质.讨论了区间值模糊格蕴涵子代数与（模糊）格蕴涵子代数之间的关系;定义了区间值模糊集的象和原象,获得了区间值模糊格蕴涵子代数的象和原象成为区间值模糊格蕴涵子代数的条件.     

模糊蕴涵代数的结构特征

本文研究了模糊蕴涵代数的一些性质,给出了模糊蕴涵代数成为Heyting代数的一个条件,得到对模糊蕴涵代数的结构特征刻画,并给出了一个(2,0)型代数(X,→,0)成为模糊蕴涵代数的充分必要条件.     

模糊蕴涵与Smooth紧

1985年,Sostak[8]由逻辑的观点出发定义了一种基于Lukasiew icz蕴涵算子的Sm ooth紧,基于同样的思想A.H aydar E s[12]又定义了另外几种Sm ooth(fuzzy)紧并讨论了它们之间的关系。王国俊教授[10]提出了一种比Lukasiew icz算子具有更好性质的R0-算子,基于R0-算子我们定义了一种模糊蕴涵,通过这种模糊蕴涵定义几种Sm ooth紧并讨论了它们的一些性质。     

格蕴涵代数的直觉模糊LI-理想

本文深入研究了格蕴涵代数的直觉模糊\textit{LI}-理想理论.给出了直觉模糊\textit{LI}-理想的若干新的性质和等价刻画. 建立了由一个直觉模糊集生成的直觉模糊\textit{LI}-理想的表示定理.证明了一个格蕴涵代数的全体之集模糊\textit{LI}-理想之集在直觉模糊包含序下构成一个完备的分配格.     

剩余型直觉模糊蕴涵算子的统一形式

对直觉三角模和直觉三角余模的性质进行研究,提出由此生成的直觉伴随对和直觉余伴随对的概念,讨论它们在直觉模糊区域上的性质,给出与直觉三角模相伴随的剩余型直觉蕴涵算子一种统一形式,最后根据直觉模糊蕴涵算子与模糊蕴涵算子的关系给出四类直觉模糊蕴涵算子的具体形式.     

正则剩余格上的模糊理想及模糊蕴涵理想

对正则剩余格的结构作进一步研究。利用正则剩余格上、算子并结合模糊数学的思想和方法,在正则剩余格上引入了模糊理想和模糊蕴涵理想的概念,讨论了它们的基本性质。主要结果是:(1)给出了模糊理想和模糊蕴涵理想的等价刻画;(2)证明了模糊蕴涵理想一定是模糊理想,模糊理想不必是模糊蕴涵理想;(3)证明了全体模糊理想之集在给定的运算下是一个完备的分配格。     

Symmetric implicational method of fuzzy reasoning

Fuzzy reasoning should take into account the factors of both the logic system and the reasoning model, thus a new fuzzy reasoning method called the symmetric implicational method is proposed, which contains the full implication inference method as its particular case. The previous full implication inference principles are improved, and unified forms of the new method are respectively established for FMP (fuzzy modus ponens) and FMT (fuzzy modus tollens) to let different fuzzy implications be used under the same way. Furthermore, reversibility properties of the new method are analyzed from some conditions that many fuzzy implications satisfy, and it is found that its reversibility properties seem fine. Lastly, the more general α-symmetric implicational method is put forward, and its unified forms are achieved.     

A survey of fuzzy implication algebras and their axiomatization

The theory of fuzzy implication algebras was proposed by Professor Wangming Wu in 1990. The present paper reviews the following two aspects of studies on FI-algebras: concepts, properties and some subclasses of FI-algebras; axiomatization of the class of FI-algebras and some of its important subclasses. The main results are summarized in the current paper, the relationships between FI-algebras and several classes of important fuzzy algebras are discussed, such as BL-algebras, MTL-algebras, and residuated lattices, and propositional calculus systems of several special classes of FI-algebras are shown.     

A new class of fuzzy implications. Axioms of fuzzy implication revisited

Many different fuzzy implication operators have been proposed; most of them fit into one of the two classes: implication operations that are based on an explicit representation of implication A → B in terms of &, , and ¬ (e.g., S-implications that are based on the formula B ¬ A), and R-implications that are based on an implicit representation of implication A → B as the weakest C for which C&B implies A. However, some fuzzy implication operations (such as b^a) cannot be naturally represented in this form. To describe such operations, we propose a new (third) class of implication operations called A-implications whose relation to &, , and ¬ is described by (implicit) axioms.     

(O,N)-蕴涵的新性质

重叠函数是一类特殊的非结合的二元聚类函数.利用重叠函数可以诱导出不同形式的模糊蕴涵.文[7]介绍了由重叠函数O和模糊否定N诱导出的模糊蕴涵—(O,N)-蕴涵,并对(O,N)-蕴涵的基本性质进行了研究,但尚存一些重要性质尚未得到研究.主要基于自同构φ和模糊否定N的相关性质对(O,N)-蕴涵进行深入研究,给出(O,N)-蕴涵一些新的重要性质.     

Implication in fuzzy logic

Intermediate truth values and the order relation “as true as” are interpreted. The material implication A → B quantifies the degree by which “B is at least as true as A.” Axioms for the → operator lead to a representation of → by the pseudo-Lukasiewicz model. A canonical scale for the truth value of a fuzzy proposition is selected such that the → operator is the Lukasiewicz operator and the negation is the classical 1−. operator. The mathematical structure of some conjunction and disjunction operators related to → are derived.     

Regularly open sets in fuzzy topological spaces

This paper is devoted to the study of the role of fuzzy regularly open sets. We prove some properties of fuzzy almost continuous mappings and define fuzzy almost open mappings. We prove that under a fuzzy almost continuous and fuzzy almost open map, the inverse image of a fuzzy regularly open set is fuzzy regularly open. Further we define a new type of fuzzy separation axioms, fuzzy almost separation axioms. It is interesting that there are some deviations in the behaviour of these axioms as compared to those in general topology. For example, in a fuzzy almost T₁ space not every fuzzy singleton is δ-closed. Also a fuzzy space which is fuzzy almost as well as fuzzy almost T₀ is fuzzy almost regular. While in general topology we have to take an almost T₂ space in place of almost T₀ space.     

Triple I method of approximate reasoning on Atanassov's intuitionistic fuzzy sets

Two basic inference models of fuzzy reasoning are fuzzy modus ponens (FMP) and fuzzy modus tollens (FMT). The Triple I method is a very important method to solve the problems of FMP and FMT. The aim of this paper is to extend the Triple I method of approximate reasoning on Atanassov's intuitionistic fuzzy sets. In the paper, we first investigate the algebra operators' properties on the lattice structure of intuitionistic fuzzy information and provide the unified form of residual implications which indicates the relationship between intuitionistic fuzzy implications and fuzzy implications. Then we present the intuitionistic fuzzy reasoning version of the Triple I principles based on the models of intuitionistic fuzzy modus ponens (IFMP) and intuitionistic fuzzy modus tollens (IFMT) and give the Triple I method of intuitionistic fuzzy reasoning for residual implications. Moreover, we discuss the reductivity of the Triple I methods for IFMP and IFMT. Finally, we propose α-Triple I method of intuitionistic fuzzy reasoning.     

单向奇异粗糙模糊集合的结构

用模糊集合与模糊等价关系对单向奇异粗集进行了研究,并给出了单向奇异粗糙模糊集合的数学结构及其并、交、补运算和性质.同时证明了单向奇异粗糙模糊集合对并、交、补运算构成完全可无限分配的软代数.