The characterization of monotone functions that generate associative functions

Consider a two-place function $T:{[0,1]}^2\to [0,1]$ defined by $T(x,y)=f^{(-1)}\left(F\right(f\left(x\right),f\left(y\right)\left)\right)$ where $F:{[0,\infty ]}^2\to [0,\infty ]$ is an associative function, $f:[0,1]\to [0,\infty ]$ is a monotone function that satisfies either $f\left(x\right)=f\left(x^{+}\right)$ when $f\left(x^{+}\right)\in \text{Ran}\left(f\right)$ or $f\left(x\right)\ne f\left(y\right)$ for any $y\ne x$ when $f\left(x^{+}\right)\notin \text{Ran}\left(f\right)$ for all $x\in [0,1]$ and $f^{(-1)}:[0,\infty ]\to [0,1]$ is a pseudo-inverse of f. In this article, the associativity of the function T is shown to depend only on properties of the range of f. The necessary and sufficient conditions for the T being associative are presented by applying the properties of the monotone function f.