The order aspect of the fuzzy real line
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.
scientific article; zbMATH DE number 3895872