On polygonal fuzzy sets and numbers
type: Item
instance of: scholarly article
title: On polygonal fuzzy sets and numbers
zbMATH Open document ID: 1260.03093
author: A. D. Báez-Sánchez
author: Antonio Carlos Moretti
author: Rojas-Medar Marko Antonio
published in: Fuzzy Sets and Systems
publication date: 2012-12-04T00:00:00Z
full work available at URL: http://hdl.handle.net/10533/130350
review text: The main purpose (and merit) of this paper is to provide a new mathematical formalization of polygonal fuzzy numbers (FNs) in dimension \(n\in \mathbb N\) which naturally includes the most famous case of trapezoidal FNs in dimension one or piecewise linear FNs in arbitrary dimension. After some preliminaries in Section \(2\), the new formalization of this notion is introduced in Definition \(5\) (Section 3), which exactly generalizes the well-known case of a one-dimensional polygonal FN, and turns out to be a relevant subclass of normal FNs on \(\mathbb R^n\). The rest of Section 3 is devoted to some properties (like completeness) of such class, heavily based upon the general Hausdorff distance between sets. In particular, the family of polygonal FNs, based upon a fixed partition \(0=\alpha_0<\alpha_1<\ldots<\alpha_m=1\) of the real unit interval, is shown to be a convex closed cone isomorphic to a particular convex closed cone of \(\mathbb R^{2(m+1)}\). The main result of this paper is given by Theorem 23, in Section \(5\), which extends to higher dimensions and general partitions of \([0,1]\) whose maximum length tends to zero. Some results already appeared when \(n=1\) and for uniform partitions, which state that an arbitrary continuous FN may be approximated by a suitable sequence of polygonal FNs using the generalized Hausdorff \(d{_\infty}\)-distance. By means of this class, the authors also show that some embedding theorems may be decisively simplified.
reviewed by: Roberto Ghiselli Ricci
Mathematics Subject Classification ID: 03E72
zbMATH DE Number: 6112232
zbMATH Keywords: fuzzy numbers
zbMATH Keywords: polygonal fuzzy numbers
zbMATH Keywords: generalized Hausdorff metric
zbMATH Keywords: approximation of fuzzy numbers
MaRDI profile type: Publication
OpenAlex ID: W2066559088
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DOI: 10.1016/J.FSS.2012.04.003
description: scientific article; zbMATH DE number 6112232