模糊商空间理论两个定理的补充

张铃等在“模糊粒度计算方法”中，核心定理证明中构造的等价关系是循环定义的，且没有证明它的截关系与商空间所对应的等价关系是相等的；模糊等价关系的粗细定义与模糊集理论的意义不一致，容易引起歧义．本文用通用的模糊数学符号和序代数理论的方法和观点对其进行修正和补充，给出两个定理的完整证明，使得相关结果的表达更简洁和规范，完善了模糊商空间理论．

Two Theorems in Fuzzy Quotient Space Theory

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.