The characterization of monotone functions that generate associative functions

Consider a two-place function [0, 1](2) -> [0, 1] defined by (, ) = ((-1))(((), ())) where [0,infinity](2) -> [0,infinity] is an associative function, [0, 1] -> [0,infinity] is a monotone function that satisfies either () = ((+)) when ((+)) is an element of Ran() or () not equal () for any not equal when (+) is not an element of Ran() for all is an element of [0, 1] and (-1) [0,infinity] -> [0, 1] is a pseudo-inverse of . In this article, the associativity of the function is shown to depend only on properties of the range of . The necessary and sufficient conditions for the being associative are presented by applying the properties of the monotone function .