A sandwich theorem for monotone additive functions
type: Item
instance of: scholarly article
title: A sandwich theorem for monotone additive functions
zbMATH Open document ID: 0834.20060
DOI: 10.1007/BF02573643
author: Peter Plappert
published in: Semigroup Forum
publication date: 1996-03-18T00:00:00Z
full work available at URL: https://eudml.org/doc/135422
review text: 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.
Mathematics Subject Classification ID: 20M14
Mathematics Subject Classification ID: 06F05
Mathematics Subject Classification ID: 20M20
Mathematics Subject Classification ID: 54H15
zbMATH DE Number: 815966
zbMATH Keywords: monotone additive functions
zbMATH Keywords: preordered Abelian monoids
zbMATH Keywords: commutative semigroups
reviewed by: jun. Kenneth D. Magill
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 815966