Monotone functions that generate conditionally cancellative triangular subnorms
Meng Chen, Xue-ping Wang
TL;DR
The paper characterizes monotone generators $f:[0,1]\to[0,1]$ for which the induced two-variable operation $F(x,y)=f^{(-1)}(T(f(x),f(y)))$ becomes a conditionally cancellative triangular subnorm, solving an open problem of Mesiarová. It establishes explicit necessary and sufficient conditions on the image $M=\text{Ran}(f)$ and related sets (e.g., $Q$, $C$) under a strict t-norm $T$, including inclusions $T(M\setminus C, M)\subseteq M\cup[0,f(0^{+})]$ and $T(Q,M)\subseteq[0,f(0^{+})]$ for conditional cancellativity; it further identifies when $F$ is cancellative or a proper t-subnorm, and provides a comprehensive framework that extends to dual t-conorms and multiplicative generators for Archimedean proper t-subnorms. The results yield a complete description of how monotone generators govern the algebraic properties of generated t-subnorms, and they resolve the open problem by Mesiarová in the continuous setting.
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 Mesiarová.