A sandwich theorem for monotone additive functions
A preordered abelian monoid is a quadruple \((S, +, 0, \prec)\) where \((S, +)\) is a commutative semigroup with neutral element 0 and the relation \(\prec\) is reflexive, transitive and compatible with addition. A mapping \(\varphi\) from \(S\) into \({\mathbf R} \cup \{\infty\}\) is said to be \(\prec\)-monotone if \(\varphi(s) \leq \varphi(t)\) whenever \(s \prec t\) where \(\mathbf R\) is the set of real numbers. The function \(\varphi\) is said to be additive if it is a homomorphism of the monoid \((S, +, 0)\) into \(({\mathbf R} \cup \{\infty\}, +, 0)\). Now let \(X\) and \(Y\) be two subsets of \(S\), let \(g\) be a function from \(X\) to \(\mathbf R\) and let \(h\) be a function from \(Y\) to \(\mathbf R\). The author investigates the problem of determining when there exists an additive \(\prec\)-monotone function \(f\) from \(S\) to \(\mathbf R\) such that \(g \leq f |X\) and \(f |Y \leq h\). In the main theorem he gives a number of conditions, some of which are necessary for the existence of such a function \(f\) and others which are both necessary and sufficient.
scientific article; zbMATH DE number 815966