When is the operation $τ_{T,L}$ a triangle function on $\Del^+$?

Abstract

This paper resolves an open problem posed by Schweizer and Sklar in 1983. We establish that the binary operation $\tauTL$ is a triangle function on $\Delp$ if and only if the following three conditions hold: (a) $L$ is a continuous t-conorm on $[0, \infty]$ satisfying $(LCS)$; (b) $T$ is a t-norm on $[0, 1]$; and (c) $T$ is weakly left continuous, with left continuity required when $L$ is non-Archimedean.