On the construction of fuzzy betweenness relations from metrics

Abstract

We consider the problem of constructing a fuzzy betweenness relation from a metric. More precisely, given a continuous Archimedean triangular norm, we present two construction methods for a fuzzy betweenness relation from a metric by making use of the pseudo-inverse of either a continuous additive generator or a continuous multiplicative generator of the triangular norm. In case the metric is bounded and given a 1-Lipschitz continuous triangular norm, we present a third construction method for a fuzzy betweenness relation from a metric by making use of the residual implication of the triangular norm. Since the Łukasiewicz and product triangular norms are both continuous Archimedean and 1-Lipschitz continuous, all three construction methods may be used. Interestingly, the construction method based on the residual implication is proved to coincide with that based on a continuous additive generator for the Łukasiewicz triangular norm and with that based on a continuous multiplicative generator for the product triangular norm. We end by noting that all three construction methods result in a fuzzy prebetweenness relation when considering a pseudometric instead of a metric.