The order aspect of the fuzzy real line
type: Item
instance of: scholarly article
title: The order aspect of the fuzzy real line
zbMATH Open document ID: 0562.54010
DOI: 10.1007/BF01215250
published in: Manuscripta Mathematica
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155046
review text: Since 1975 several different fuzzy real lines have been actively studied. These include Hutton's line [\textit{B. Hutton}, J. Math. Anal. Appl. 50, 74-79 (1975; Zbl 0297.54003)], Höhle's line [\textit{U. Höhle}, Manuscripta Math. 26, 223-245 (1978; Zbl 0413.54003)], and the saturated Hutton line. The present paper makes a valuable contribution because it provides a unifying framework for these fuzzy lines when the underlying lattice is the unit interval. In addition the author identifies the Hutton lines in a probabilistic context, which is closer to the central concerns of most researchers in fuzzy sets. Previously the author has shown [Math. Nachr. 115, 33-57 (1984; Zbl 0593.54006)] that each subbasis of a metrizable topology for X generates a fuzzy topology on the set of Borel probability measures, \({\mathcal M}(X)\). Applying this result to \({\mathbb{R}}\), he first shows that the three fuzzy lines are canonically homeomorphic to spaces with the same underlying set, \({\mathcal M}({\mathbb{R}})\). The three topologies are appropriate variations of Lowen's construction and the same map is the homeomorphism in each case. The result also holds for fuzzy intervals. The second part of the paper introduces a fuzzy relation \(\rho\) on \({\mathcal M}({\mathbb{R}})\) which extends the usual order, \(<\). Moreover, \(\rho\) induces the topologies of the Höhle and saturated Hutton lines in a manner analogous to the way that \(<\) induces the topology of \({\mathbb{R}}\). The unsaturated topology induced by \(\rho\) is strictly finer than the Hutton topology, a fact that creates the possibility of a fourth fuzzy real line. On a more philosophical level, the author argues that the order relation and not the topology should be the crucial determinant of a fuzzy real line.
Mathematics Subject Classification ID: 54A40
Mathematics Subject Classification ID: 60B05
zbMATH DE Number: 3895872
zbMATH Keywords: fuzzy real lines
zbMATH Keywords: Hutton's line
zbMATH Keywords: Höhle's line
zbMATH Keywords: saturated Hutton line
zbMATH Keywords: fuzzy relation
author: Robert Lowen
MaRDI profile type: Publication
cites work: Probabilistische Topologien
cites work: Probabilistische Metriken auf der Menge der nicht negativen Verteilungsfunktionen
cites work: Probabilistisch kompakte L-unscharfe Mengen
cites work: Normality in fuzzy topological spaces
cites work: Uniformities on fuzzy topological spaces
cites work: Ordered fuzzy topological spaces
cites work: Generalizing the L-fuzzy unit interval
cites work: Construction of fuzzy sigma-algebras using triangular norms
cites work: Fuzzy topological spaces and fuzzy compactness
cites work: Initial and final fuzzy topologies and the fuzzy Tychonoff theorem
cites work: A comparison of different compactness notions in fuzzy topological spaces
cites work: On the Existence of Natural Fuzzy Topologies on Spaces of Probability Measures
cites work: Q3684801
cites work: A complete characterization of the level spaces of R(I) and I(I)
cites work: On (R(L),+)
cites work: Concerning the constants in fuzzy topology
cites work: Q4693151
cites work: Q5509983
cites work: Q5533878
cites work: Fuzzy addition in the L-fuzzy real line
cites work: A categorical accommodation of various notions of fuzzy topology
Recommended article: On the Existence of Natural Fuzzy Topologies on Spaces of Probability Measures
Recommended article: Basic notions for fuzzy topology. I
Recommended article: Compactness properties in the fuzzy real line
Recommended article: Q3684801
Recommended article: Generalizing the L-fuzzy unit interval
Recommended article: Fuzzy real numbers
Recommended article: Fuzzy topology generated by fuzzy metrics
Recommended article: Fuzzy real line structure and metric space
Recommended article: Connectedness on fuzzy topological spaces
Recommended article: Dynamic topologies and their applications to crisp topologies, fuzzifications of crisp topologies, and fuzzy topologies on the crisp real line
description: scientific article; zbMATH DE number 3895872