Characterizations of monotone right continuous functions which generate associative functions

Associativity of a two-place function $T:[0,1{]}^2\to [0,1]$ defined by $T(x,y)=f^{(-1)}(T^{\ast }(f(x),f(y)))$ where $T^{\ast }:[0,1{]}^2\to [0,1]$ is an associative function with neutral element in $[0,1]$, $f:[0,1]\to [0,1]$ is a monotone right continuous function and $f^{(-1)}:[0,1]\to [0,1]$ is the 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 right continuous function $f$.