Characterizations of monotone right continuous functions which generate associative functions

Associativity of a two-place function T : [0, 1](2)->[0, 1] defined by T(x, y) = f((-1))(T*(f(x), f(y))) where T* : [0, 1](2) -> [0, 1] is an associative function with neutral element in [0, 1], f : [0, 1] -> [0, 1] is a monotone right continuous function and f((-1)) : [0, 1] -> [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.