模糊集范畴  

Categories of Fuzzy Sets

作　　者：张德学[1] ZHANG Dexue(College of Mathematics,Sichuan University,Chengdu 610064,Sichuan)

机构地区：[1]四川大学数学学院,四川成都610064

出　　处：《四川师范大学学报（自然科学版）》2024年第1期17-24,共8页Journal of Sichuan Normal University（Natural Science）

基　　金：国家自然科学基金(12371463)。

摘　　要：在模糊集的理论和应用研究中,一个基本问题是怎样把模糊集组织成一个范畴,相应的范畴具有什么样的性质?这一问题自Zadeh引入模糊集的概念之后,一直受到人们的关注.简要介绍2个具有代表性的解决方案,其一是由Goguen提出的Goguen范畴,其二是受topos理论影响发展起来的Q-集理论.这2种方案差异很大,Goguen范畴把模糊集的隶属函数看作一个集合到单点集的模糊关系,因而2个模糊集之间的态射就是2个集合之间满足一定条件的模糊关系;Q-集范畴则把隶属函数看作类型函数,因而2个模糊集之间的态射就是2个类型函数之间满足一定条件的模糊关系.这2种方案的共同特点在于充分利用逻辑值域的逻辑结构,这也是模糊集理论区别于其他数学分支的一个基本特征.A basic problem in theoretic or application-oriented investigations of fuzzy sets is how to organize fuzzy sets into a category,and what properties does the resulting category posses?This problem has received wide attention ever since the introduction of the fuzzy set by Zadeh.This paper presents a brief introduction to two solutions.The first is the Goguen category proposed by Goguen,and the second is the theory of Q-sets motivated by topos theory.In the Goguen category,the membership function of a fuzzy set is viewed as a fuzzy relation between a set and the singleton set,so a morphism between two fuzzy sets is defined to a fuzzy relation between sets satisfying certain requirement.In the theory of Q-sets,the membership function is viewed as a type function,and a morphism between two fuzzy sets is then a fuzzy relation between the fuzzy sets subject to certain requirement.These two solutions are quite different,where the common point is to make full use of the logic structures of the truth-value set,which is a distinctive feature of mathematical theories of fuzzy sets.

关 键 词：模糊集 QUANTALE Goguen范畴 强化范畴 

分 类 号：O159[理学—数学] O154[理学—基础数学]