A New Approach to the Fuzzification of Convex Structures

Abstract

A new approach to the fuzzification of convex structures is introduced. It is also called an $M$-fuzzifying convex structure. In the definition of $M$-fuzzifying convex structure, each subset can be regarded as a convex set to some degree. An $M$-fuzzifying convex structure can be characterized by means of its $M$-fuzzifying closure operator. An $M$-fuzzifying convex structure and its $M$-fuzzifying closure operator are one-to-one corresponding. The concepts of $M$-fuzzifying convexity preserving functions, substructures, disjoint sums, bases, subbases, joins, product, and quotient structures are presented and their fundamental properties are obtained in $M$-fuzzifying convex structure.