On the generalized associativity equation

The so-called generalized associativity functional equation G(J(x,y),z)=H(x,K(y,z))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} G(J(x,y),z) = H(x,K(y,z)) \end{aligned}$$\end{document}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))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} F(x,y,z) = G(J(x,y),z) = H(x,K(y,z)) \end{aligned}$$\end{document}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.