Let \((X_1, {\operatorname {d}}_{X_1}), \ldots, (X_n, {\operatorname {d}}_{X_n})\) be metric spaces. Let \(\bar {x}=(x_1,\ldots, x_n)\) for \(x_i\in X_i\) and \(\bar {D}(\bar {x},\bar {y})=\bigl ({\operatorname {d}}_{X_1}(x_1,y_1),\ldots {\operatorname {d}}_{X_n}(x_n,y_n)\bigr)\). A metric \(d\) on \(X_1\times \cdots \times X_n\) is a metric product if \(d(\bar {x},\bar {y})=\Phi \bigl (\bar {D}(\bar {x},\bar {y})\bigr)\) for some function \(\Phi \:{\mathbb R}^n_+\to {\mathbb R}_+\). A function \(\Phi \:{\mathbb R}^n_+\to {\mathbb R}_+=[0,\infty)\) is a metric preserving function if \(\Phi \circ \bar {D}\) is a metric on \(X_1\times \cdots \times X_n\) for every collection of metric spaces \((X_1, {\operatorname {d}}_{X_1}), \ldots, (X_n, {\operatorname {d}}_{X_n})\). For \(A\subset {\mathbb R}^n_+\), a function \(f\: A\to {\mathbb R}_+\) is isotone if \(f(\bar {x})\leq f(\bar {y})\) whenever \(\bar {x}\leq \bar {y}\) (\(\bar {x}\leq \bar {y}\) if \(x_i\leq y_i\) for \(i=1,\ldots, n\)). In the paper, isotone metric preserving functions are investigated.
Typical result: Let \((X_1, {\operatorname {d}}_{X_1}), \dots, (X_n, {\operatorname {d}}_{X_n})\) be nonempty metric spaces. Then a metric \(d\) on \(X_1\times \dots \times X_n\) is an isotone subadditive metric product if and only if there is an isotone metric preserving function \(\Psi \:{\mathbb R}^n_+\to {\mathbb R}_+\) such that \(d(\bar {x},\bar {y})=\Psi \bigl (\bar {D}(\bar {x},\bar {y})\bigr)\) for all \(\bar {x},\bar {y}\in X_1\times \cdots \times X_n\).