tailieunhanh - Interpolative relations and interpolative preference structures

Relations are very important mathematical objects in different fields of theory and applications. In many real applications, for which gradation of relations is immanent, the classical relations are not adequate. Interpolative relations (I-relations) (as fuzzy relations) are the generalization of classical relations so that the value (intensity) of a relation is an element from a real interval [0, 1] and not only from {0, 1} as in the classical case. | Yugoslav Journal of Operations Research 15 (2005), Number 2, 171-189 INTERPOLATIVE RELATIONS AND INTERPOLATIVE PREFERENCE STRUCTURES Dragan RADOJEVIĆ Mihajlo Pupin Institute Belgrade, Serbia & Montenegro Received: February 2005 / Accepted: July 2005 Abstract: Relations are very important mathematical objects in different fields of theory and applications. In many real applications, for which gradation of relations is immanent, the classical relations are not adequate. Interpolative relations (I-relations) (as fuzzy relations) are the generalization of classical relations so that the value (intensity) of a relation is an element from a real interval [0, 1] and not only from {0, 1} as in the classical case. The theory of I-relations is crucially different from the theory of fuzzy relations. I-relations are consistent generalizations of classical relations and, contrary to fuzzy relations, all laws of classical relations (set-theoretical laws) are preserved in general case. In this paper, the main characteristics of I-relations are illustrated on the interpolative preference structures (Ipreference structures) as consistent generalization of classical preference structures. Keywords: Fuzzy relations, interpolative relations (I-relations), symbolic level of I-relations, structure of I-relations, primary, atomic and combined I-relations, valued level of I-relations, intensity of I-relations, generalized product, interpolative preference (I-preference) structure. 1. INTRODUCTION Classical relations based on classical logic and/or classical set theory are very useful in classical mathematics and almost all applications for which “black & white” approach is appropriate. In many real applications the classical relations are not adequate. This was the motive for development of fuzzy relations [4]. Fuzzy relations are based on fuzzy logic and/or theory of fuzzy sets [8]. Fuzzy logics are truth functional. Logic is truth functional if .