First, a new definition of conjugate mapping concept for convex fuzzy mapping is given in this paper, which is more reasonable than the concept in the literature. Then, we prove that the secondary conjugate mapping of convex fuzzy mapping is convex fuzzy mapping and that the relationship between two convex fuzzy mappings and their intimal convolution's conjugate mappings. Besides, the definition of conjugate mapping for general fuzzy mappings is given, which is the extension of the concept of conjugate mapping of convex fuzzy mapping. Moreover, we prove that conjugate mapping is convex mapping and that secondary conjugate mapping is convex fuzzy mapping. Finally, we discuss the relationship between the sub-differential of fuzzy mapping and conjugate mapping and prove some relational expressions.
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