Solution of generalized bisymmetry type equations without surjectivity assumptions
type: Item
instance of: scholarly article
title: Solution of generalized bisymmetry type equations without surjectivity assumptions
zbMATH Open document ID: 0924.39014
published in: Aequationes Mathematicae
publication date: 1999-11-07T00:00:00Z
review text: The \(m\times n\) generalized bisymmetry equation (all functions unknown) \[ F[G_{1}(x_{11},\ldots,x_{1n}),\ldots, G_{m}(x_{m1},\ldots,x_{mn})]= G[F_{1}(x_{11},\ldots,x_{m1}), \ldots,F_{n}(x_{1n},\ldots,x_{mn})] \] characterizes consistent aggregation in mathematical economics. It has a long history [see e.g. the surveys \textit{J. van Daal} and \textit{A. H. Q. M. Merkies} in Measurement in Economics, ed. W. Eichhorn, Physica, Heidelberg, 607-637 (1987) and \textit{J. Aczél} in Choice, Decision, and Measurement, ed. A. A. J. Marley, Lawrence Erlbaum Ass. Publ., Mahwah, NJ, 225-233 (1997)]. Recent results assume ``surjectivity'' conditions of the following type (not spelled out in this paper): holding all but one variable constant the (unknown) function still assumes all values it would without fixing any of the variables. The present paper disposes of such conditions on Cartesian product of nondegenerate real intervals, while assuming continuity and strict monotonicity in each variable. The proof is difficult. It not only applies the usual double induction but has to piece together solutions on subintervals. However, the usual result is obtained: all unknown functions are quasi-sums (i.e. continuous isotopes of sums).
reviewed by: János Aczél
Mathematics Subject Classification ID: 39B22
Mathematics Subject Classification ID: 91B38
Mathematics Subject Classification ID: 91B60
zbMATH DE Number: 1295530
zbMATH Keywords: generalized bisymmetry
zbMATH Keywords: functional equations
zbMATH Keywords: quasisums
zbMATH Keywords: consistent aggregation
zbMATH Keywords: mathematical economics
author: Gyula Maksa
MaRDI profile type: Publication
full work available at URL: s000100050070
OpenAlex ID: W2002233314
DOI: 10.1007/S000100050070
description: scientific article; zbMATH DE number 1295530