Quasisums and generalized associativity
The generalized associativity equation \[ F(G(x,y),z)=H(x,K(y,z)) \tag{GA} \] has been studied by several authors in various settings. In particular Aczél, Belousov and Hosszú solved (GA) on quasigroups. Due to quasigroup techniques surjectivity assumptions were made. The author of the present paper already studied (GA) with the aim of eliminating surjectivity conditions. In the paper under review he presents a possible way to find the solutions of (GA) that are continuous and strictly monotone in each variable. A real--valued function is called a \(CM\) function if it is defined on a real interval or on a rectangle in the real plane (with sides parallel to the axes) and it is continuous and strictly monotone in each variable. The main result of the paper is the following: Theorem: Let \(X\), \(Y\) and \(Z\) be intervals, \(G:X \times Y \to \mathbb R\), \(K:Y \times Z \to \mathbb R\), \(F:G(X,Y) \times Z \to \mathbb R\) and \(H:X \times K(Y,Z) \to \mathbb R\) be \(CM\) functions. Then the generalized associativity equation \[ F(G(x,y),z)=H(x,K(y,z)) \tag{GA} \] holds for all \((x,y,z) \in X \times Y \times Z\), if and only if there exist \(CM\) functions \(\alpha:X \to \mathbb R\), \(\beta:Y \to \mathbb R\), \(\gamma:Z \to \mathbb R\), \(\delta_1:\alpha(X)+\beta(Y) \to \mathbb R\), \(\delta_2:\beta(Y)+\gamma(Z) \to \mathbb R\) and \(\phi:\alpha(X)+\beta(Y)+\gamma(Z) \to \mathbb R\) such that \[ \begin{aligned} F(\xi,z)&=\phi(\delta_1^{-1}(\xi)+\gamma(z)),\\ G(x,y)&=\delta_1(\alpha(x)+\beta(y)),\\ H(x,\eta)&=\phi(\alpha(x)+\delta_2^{-1}(\eta)),\\ K(y,z)&=\delta_2(\beta(y)+\gamma(z)),\\ \end{aligned} \] hold for all \(x \in X\), \(y \in Y\), \(z \in Z\), \(\xi \in G(X,Y)\) and \(\eta \in K(Y,Z)\).
scientific article; zbMATH DE number 2162495