Affiliations: [a] INAMAT and Departamento de Matemáticas, Universidad Pública de Navarra, E-31006 Pamplona, Spain | [b] Departamento de Matemáticas, Universidad de Zaragoza, E-50009 Zaragoza, Spain
Correspondence:
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Corresponding author. Esteban Induráin, INAMAT and Departamento de Matemáticas, Universidad Pública de Navarra, E-31006 Pamplona, Spain. Tel.: +34 948169551; Fax: +34 948166057; E-mail: steiner@unavarra.es
Note: [1] This work has been supported by the research projects MTM2012-37894-C02-02 and TIN2013-40765-P (Spain). Thanks are given to Prof. Dr. G. Ochoa (Pamplona, Spain) for his valuable suggestions and comments
Abstract: 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.
Keywords: Functional equations in two variables, fuzzy sets, representable total preorders, interval orders and semiorders, separability functional equation, AMS Subject Classification (2010): 39B522 (Primary), 03E72, 06A06, 39B22
