Metrics and T-equalities

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