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

TL;DR

This work resolves the Schweizer–Sklar open problem by characterizing exactly when the binary operation $\tau_{T,L}$ on $\Delta^+$ forms a triangle function. It proves necessity of three conditions on $T$ and $L$ (continuous t-conorm $L$ on $[0,\infty]$ with $(LCS)$, t-norm $T$ on $[0,1]$, and weakly left continuous $T$ with left continuity when $L$ is non-Archimedean) and provides a nontrivial sufficiency proof that hinges on associativity for the Archimedean and non-Archimedean cases. The sufficiency part, including a technical associativity argument, completes the construction of continuous triangle functions on $\Delta^+$ from given $T$ and $L$, thereby fully solving the problem posed by Schweizer and Sklar. The results advance probabilistic metric space theory by clarifying when probabilistic distance operations behave like triangle functions.

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.

Theorems & Definitions (26)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Proposition 2.5
• proof
• Proposition 2.6
• Proposition 2.7
• Corollary 2.8
• Corollary 2.9