Monotone functions that generate conditionally cancellative triangular subnorms

Let a function $F:[0,1{]}^2\to [0,1]$ be given by $F(x,y)=f^{(-1)}(T(f(x),f(y)))$ where $f:[0,1]\to [0,1]$ 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.