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

TL;DR

This paper characterizes when the operation _{T,L} is a triangle function on ^+ by establishing necessary and sufficient conditions involving t-norms and t-conorms.

Contribution

It provides a complete characterization of the conditions under which _{T,L} forms a triangle function, resolving an open problem from 1983.

Findings

_{T,L} is a triangle function if and only if L is a continuous t-conorm satisfying (LCS)

T must be a t-norm on [0,1]

T must be weakly left continuous, with additional continuity when L is non-Archimedean

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.