The characterizations of monotone functions which generate associative functions

Associativity of a two-place function $T:[0,1{]}^2\to [0,1]$ defined by $T(x,y)=f^{(-1)}(F(f(x),f(y)))$ where $F:[0,\mathrm{\infty }{]}^2\to [0,\mathrm{\infty }]$ is an associative function, $f:[0,1]\to [0,\mathrm{\infty }]$ is a monotone function which satisfies either $f(x)=f(x^{+})$ when $f(x^{+})\in Ran(f)$ or $f(x)\mathrm{\ne }f(y)$ for any $y\mathrm{\ne }x$ when $f(x^{+})\mathrm{\notin }Ran(f)$ for all $x\in [0,1]$ and $f^{(-1)}:[0,\mathrm{\infty }]\to [0,1]$ is a pseudo-inverse of $f$ depends only on properties of the range of $f$. The necessary and sufficient conditions for the $T$ to be associative are presented by applying the properties of the monotone function $f$.