Metrics and T-equalities

Abstract

The relationship between metrics and F-equalities is investigated; the latter are a special case of F-equivalences, a natural generalization of the classical concept of an equivalence relation. It is shown that in the construction of metrics from F-equalities triangular norms with an additive generator play a key role. Conversely, in the construction of F-equalities from metrics this role is played by triangular norms with a continuous additive generator or, equivalently, by continuous Archimedean triangular norms. These results are then applied to the biresidual operator ℰF of a triangular norm F. It is shown that ℰF is a F-equality on [0, 1] if and only if F is left-continuous. Furthermore, it is shown that to any left-continuous triangular norm F there correspond two particular F-equalities on F(X), the class of fuzzy sets in a given universe X; one of these F-equalities is obtained from the biresidual operator ℰF by means of a natural extension procedure. These F-equalities then give rise to interesting metrics on F(X). © 2002 Elsevier Science (USA).