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

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 φ-transform of the Łukasiewicz triangular norm. In this second part, additional arguments for this statement are supplied in what we call the 'identity case' (φ_τ 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, 4/5]² it actually behaves as the Łukasiewicz 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 Łukasiewicz triplet. These additional arguments conclude in a convincing way our plea for the use of Łukasiewicz triplets in the definition of fuzzy preference structures.