A Characterization of Convex Fuzzy Mappings

A Characterization of Convex Fuzzy Mappings

Chih-Yuan Chen (Nanya Institute of Technology, Taiwan), Cheng-Pin Wang (Yuan-Ze University, Taiwan) and Tetz C. Huang (Yuan-Ze University, Taiwan)
Copyright: © 2010 |Pages: 5
DOI: 10.4018/jalr.2010070104

Abstract

As any real-valued functions can be regarded as a fuzzy mapping, the research on fuzzy mappings extends the research on real-valued functions. In this paper, a characterization of convex fuzzy mappings is obtained. Supposeis a fuzzy mapping, whereis a non-empty convex subset ofandis the set of all fuzzy numbers. With respect to the fuzzy-max order, is convex if and only if it is both quasi-convex and intermediate-point convex.
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Preliminaries

Letjalr.2010070104.m07denote the jalr.2010070104.m08-dimensional Euclidean space. A fuzzy set ofjalr.2010070104.m09is defined to be a function fromjalr.2010070104.m10intojalr.2010070104.m11. The support,jalr.2010070104.m12, of a fuzzy setjalr.2010070104.m13is defined by:

jalr.2010070104.m14.

Forjalr.2010070104.m15, the jalr.2010070104.m16-level setjalr.2010070104.m17of a fuzzy setjalr.2010070104.m18is defined by:

jalr.2010070104.m19
wherejalr.2010070104.m20denotes the closure ofjalr.2010070104.m21. A fuzzy number considered in this paper is a fuzzy setjalr.2010070104.m22with the following properties:

  • (1)jalr.2010070104.m23is normal, i.e.,jalr.2010070104.m24,

  • (2)jalr.2010070104.m25is fuzzy convex, i.e.,jalr.2010070104.m26, for anyjalr.2010070104.m27and for anyjalr.2010070104.m28,

  • (3)jalr.2010070104.m29is upper semicontinuous, i.e., for anyjalr.2010070104.m30,jalr.2010070104.m31is a closed subset ofjalr.2010070104.m32, and

  • (4)jalr.2010070104.m33is bounded.

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