On the generalized associativity equation

TL;DR

This paper explores the solutions to a generalized associativity functional equation, focusing on characterizing functions that can be decomposed into specific compositions involving given functions J and K.

Contribution

It provides a method to find all functions F that can be expressed via the generalized associativity equation given functions J and K, especially when these functions have certain range properties.

Findings

Characterization of functions F satisfying the generalized associativity equation

Solution method when J or K has the same range as one of its sections

Extension of previous work on functional equations with monotonicity assumptions

Abstract

The so-called generalized associativity functional equation G(J(x,y),z) = H(x,K(y,z)) has been investigated under various assumptions, for instance when the unknown functions G, H, J, and K are real, continuous, and strictly monotonic in each variable. In this note we investigate the following related problem: given the functions J and K, find every function F that can be written in the form F(x,y,z) = G(J(x,y),z) = H(x,K(y,z)) for some functions G and H. We show how this problem can be solved when any of the inner functions J and K has the same range as one of its sections.