Metrics and -Equalities

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