Monotone functions that generate conditionally cancellative triangular subnorms

TL;DR

This paper characterizes specific monotone functions that generate conditionally cancellative triangular subnorms, providing a complete solution to an open problem in the theory of triangular norms and subnorms.

Contribution

It offers a complete characterization of monotone functions producing conditionally cancellative triangular subnorms, resolving an open problem posed by Mesiarová.

Findings

Characterization of monotone functions generating conditionally cancellative triangular subnorms

Resolution of an open problem in the theory of triangular norms

Advancement in understanding the structure of subnorms and their generating functions

Abstract

Let a function $F: [0,1]^2\rightarrow [0,1]$ be given by $F(x,y)= f^{(-1)}(T(f(x), f(y)))$ where $f :[0,1]\rightarrow [0,1]$ is a monotone function, $f^{(-1)}$ is the pseudo-inverse of $f$ and $T$ is a triangular norm. This article characterizes the monotone function $f$ satisfying that the function $F$ is a conditionally cancellative triangular subnorm completely. It finally answers an open problem posed by Mesiarov\'{a}.