A plea for the use of Łukasiewicz triplets in the definition of fuzzy preference structures. (II). The identity case

✍ Scribed by Bernard De Baets; Bartel Van de Walle; Etienne Kerre

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 509 KB
• Volume: 99
• Category: Article
• ISSN: 0165-0114
• DOI: 10.1016/s0165-0114(96)00396-x

✦ Synopsis

It has been shown in the first part of this paper that the concept of a fuzzy preference structure is only meaningful provided that the de Morgan triplet involved contains a continuous Archimedean triangular norm having zero divisors, or hence a q~-transform of the Lukasiewicz triangular norm. In this second part, additional arguments for this statement are supplied in what we call the 'identity case' (q~ is the identity mapping, and the involutive negator is the standard negator). First, it is shown that the use of a continuous non-Archimedean triangular norm having zero divisors in the definition of a fuzzy preference structure indeed is possible. Secondly, using such a triangular norm implies that at least in the square [0, ~]2 it actually behaves as the Lukasiewicz triangular norm. Furthermore, an important transformation theorem indicates that any fuzzy preference structure with respect to a de Morgan triplet containing such a continuous non-Archimedean triangular norm having zero divisors can be transformed into a fuzzy preference structure with respect to the standard Lukasiewicz triplet. These additional arguments conclude in a convincing way our plea for the use of Lukasiewicz triplets in the definition of fuzzy preference structures.