Functional equations related to fuzzy sets and representable orderings1

Weightable quasi-metrics related to fuzzy sets

“…Finally, another unexplored possible application follows from the relationship between fuzzy sets and representable total preorders (see [5]). Having in mind studies on extensions of representable total preorders from a set to a bigger set (see e.g.…”

“…Thus, the concept of a fuzzy set can actually be regarded from various alternative points of view, as functional equations (see [6,5]), nested families of sets and nested topologies (see [1,7,20]) or representable total preorders (see [12,6]) among others. In addition, in [8] the concept of a weightable quasi-metric was shown to be closely related to some classical functional equations, of which some were also intimately linked to the denition of a fuzzy set of a universe as proved in [6,5]. However, to complete the panorama, a direct link between the denition of a fuzzy set of a universe and the possibility of endowing that universe with a suitable weightable quasi-metric had not been explored yet.…”