The restricted hull operator of M-fuzzifying convex structures1

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In this paper, the notion of M-fuzzifying restricted hull operator was defined and several equivalent characterizations were given. The notion of M-fuzzifying betweennesses was also proposed. As applications, some properties of the cut convex structures of an M-fuzzifying convex structure and of M-fuzzifying convexity preserving functions and of M-fuzzifying convex-to-convex functions were derived. As we have seen, an M-fuzzifying convex structure (X, C) is determined by its M-fuzzifying restricted hull operator h. Therefore, one can use M-fuzzifying restricted hull operators to construct M-fuzzifying convexities, as shown in Example 4.10. From the viewpoint of theory generalization, there are many work to do. For instance, by Remark 4.7, the M-fuzzifying convexity C induced by an M-fuzzy partial order e is a reasonable generalization of the order convexity of a partially ordered set. As the order convexity of a partially ordered set is JHC, S3 and its segment operator is geometric, we need to consider how to generalize these properties to M-fuzzifying convexity C.

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