A new approach to the fuzzification of convex structures is introduced. It is also called anM-fuzzifying convex structure. In the definition ofM-fuzzifying convex structure, each subset can be regarded as a convex set to some degree. AnM-fuzzifying convex structure can be characterized by means of itsM-fuzzifying closure operator. AnM-fuzzifying convex structure and itsM-fuzzifying closure operator are one-to-one corresponding. The concepts ofM-fuzzifying convexity preserving functions, substructures, disjoint sums, bases, subbases, joins, product, and quotient structures are presented and their fundamental properties are obtained inM-fuzzifying convex structure.
In this paper, the notion of (L, M)-fuzzy convex structures is introduced. It is a generalization of L-convex structures and M-fuzzifying convex structures. In our definition of (L, M)-fuzzy convex structures, each L-fuzzy subset can be regarded as an L-convex set to some degree. The notion of convexity preserving functions is also generalized to lattice-valued case. Moreover, under the framework of (L, M)-fuzzy convex structures, the concepts of quotient structures, substructures and products are presented and their fundamental properties are discussed. Finally, we create a functor ω from MYCS to LMCS and show that MYCS can be embedded in LMCS as a coreflective subcategory, where MYCS and LMCS denote the category of M-fuzzifying convex structures and the category of (L, M)-fuzzy convex structures, respectively.
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