A PLEA FOR THE USE OF LUKASIEWICZ TRIPLETS IN THE DEFINITION OF FUZZYPREFERENCE STRUCTURES - (I) - GENERAL ARGUMENTATION

The generalization of the concept of a classical (or crisp) preference structure to that of a fuzzy preference structure, expressing degrees of strict preference, indifference and incomparability among a set of alternatives, requires the choice of a de Morgan triplet, i.e., of a triangular norm and an involutive negator. The resulting concept is on ly meaningful provided that this choice allows the representation of t ruly fuzzy preferences. More specifically, one of the degrees of stric t preference, indifference or incomparability should always be unconst rained to the preference modeller. This intuitive requirement is viola ted when choosing a triangular norm without zero divisors, since in th at case fuzzy preference structures reduce to classical preference str uctures, and hence none of the degrees can be freely assigned. Further more, it is shown that the choice of a continuous non-Archimedean tria ngular norm having zero divisors is not compatible with our basic requ irement: the sets of degrees of strict preference, indifference and in comparability in [0, 1] are always bounded from above by a value stric tly smaller than i. These fundamental results imply that when working with a continuous triangular norm, only Archimedean ones having zero d ivisors are suitable candidates. These arguments sufficiently support our plea for the use of Lukasiewicz triplets in the definition of fuzz y preference structures. (C) 1998 Published by Elsevier Science B.V. A ll rights reserved.