International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems Vol. 17, No. 2 (2009) 197–219 c  World Scientific Publishing Company UNIVERSAL CODOMAINS TO REPRESENT INTERVAL ORDERS JUAN CARLOS CANDEAL Depto. de An´alisis Econ´omico, Facultad de Ciencias Econ´ omicas y Empresariales, Universidad de Zaragoza. c/ Doctor Cerrada 1-3, 50005 Zaragoza, Spain candeal@unizar.es JAVIER GUTI ´ ERREZ GARC ´ IA Depto. de Matem´aticas, Universidad del Pa´ ıs Vasco-Euskal Herriko Unibertsitatea, Apdo. 644, 48080 Bilbao, Spain javier.gutierrezgarcia@ehu.es ESTEBAN INDUR ´ AIN * Depto. de Matem´aticas, Universidad P´ ublica de Navarra, Campus Arrosad´ ıa, 31006 Pamplona, Spain steiner@unavarra.es Received 27 February 2008 Revised 10 October 2008 Given a binary relation defined on a set, we study its representability by means of a monotonic function that takes values on a suitable universal codomain (that depends on the kind of relation considered). We pay an special attention to the representability of interval orders, studying their alternative universal codomains, some of them equivalent to the set of symmetric triangular fuzzy numbers. Keywords : Interval orders; fuzzy numbers; universal codomains; continuous representa- tions of orderings. 1. Introduction In the present paper, we address the problem of the representability of orderings defined on a set X . Among the different kinds of representability, it is plain that the most classical one is the numerical representability through real-valued functions. In this framework, initiated by Cantor in 1895, the mathematical statement of the underlying problem is simple: 1–3 We are given a nonempty set X endowed with a binary relation R. We are looking for functions F : X → R accomplishing that * Corresponding author. 197 Int. J. Unc. Fuzz. Knowl. Based Syst. 2009.17:197-219. Downloaded from www.worldscientific.com by UNIVERSIDAD DEL PAIS VASCO BIBLIOTECA on 11/29/12. For personal use only.