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 imminent, 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 (I-preference structures) as consistent generalization of classical preference structures.