Functional equations related to fuzzy sets and representable orderings

The original definition of a fuzzy set hides several additional structures related to functional equations. In the present paper, we analyze the separability functional equation and some of its variants, proving that the solutions of those equations can be described in terms of pairs of linked fuzzy sets on the same universe. In addition, we show that representable total preorders, interval orders and semiorders on a universe can also be analyzed throughout separability functional equations and, consequently, by means of fuzzy sets. Other functional equations related to fuzzy sets, as well as some relationship between this setting and other approaches as the construction of quasi-metrics on a universe are also pointed out as a by-product.