Two Theorems in Fuzzy Quotient Space Theory
Mingfen Wu, Ting Shen, Cungen Cao, Wu Xianwei · Journal of Beijing Jiaotong University · 2009
Zhang Bo and Zhang Ling,in their paper Theory of fuzzy quotient space,proposed a new method of granular computing which combine fuzzy sets and theory of quotient space.Unfortunately,the proof of two core theorems was not tight and complete enough,i.e.,the fuzzy equivalence relation constructed in the proof process of theorem,is cyclical definition,and its cut relations and quotient space corresponding to the equivalence relation are equal which isn't obvious.Moreover,the definition of inclusion for fuzzy equivalence relations is not inconsistent with the significance of the fuzzy set theory,easily lead to ambiguity.In this paper,with the fuzzy mathematical symbols and the method of general order algebraic theory,the proof of two core theorems is amended,and given complete proof.So making expression of relevant results and norms are more concise and standard,and perfect the theory of fuzzy quotient space.