Abstract
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.
Highlights
Introduction and PreliminariesConvexity theory has been accepted to be of increasing importance in recent years in the study of extremum problems in many areas of applied mathematics
In 1994, Rosa presented the notion of fuzzy convex structures in [3, 4]
By Remark 16 and Theorems 13 and 29, we can obtain the following theorem, which shows that an M-fuzzifying convex structure can be characterized by means of an M-fuzzifying closure operator, which satisfies (CL0)–(CL3) and (MDF)
Summary
Convexity theory has been accepted to be of increasing importance in recent years in the study of extremum problems in many areas of applied mathematics. The concept of convexity which was mainly defined and studied in Rn in the pioneering works of Newton, Minkowski, and others as described in [1] finds a place in several other mathematical structures such as vector spaces, posets, lattices, metric spaces, graphs, and median algebras. A subset C of 2X is called a convexity if it satisfies the following three conditions:. For a nonempty set X and a subset C of MX, (X, C) is called a fuzzy convex structure if and only if (X, C) satisfies the following conditions:. A mapping C : 2X → M is called an M-fuzzifying closure system if it satisfies the following conditions:. An M-fuzzifying closure operator on X is a mapping cl : 2X → MX satisfying the following conditions:. If (V, μ) is a fuzzy vector space, for each a ∈
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