A characterization of sets and operations invariant under bijections
type: Item
instance of: scholarly article
title: A characterization of sets and operations invariant under bijections
zbMATH Open document ID: 1055.39035
author: Lech Bartłomiejczyk
author: Józef Drewniak
published in: Aequationes Mathematicae
publication date: 2004-11-05T00:00:00Z
review text: The authors call an operation (here binary, in the paper mostly \(n\)-ary) \(F: [0,1]\times[0,1]\to[0,1]\) invariant (or selfconjugate) if, for every bijection \(f\) from \([0,1]\) onto \([0,1]\) the equation \(F[f(x),f(y)]=f[F(x,y)]\) \((x,y\in[0,1])\) holds. The following is one result (the last) and the end of the paper. There exist exactly two increasing (i.e. nondecreasing in each variable) invariant binary operations: \textit{min} and the one with \(F(x,1)=x\), \(F(1,y)=y\) and \(F\) zero everywhere else on \([0,1]^2.\) ``More detailed considerations of this rich family of invariant binary operations are left to other papers''.
Mathematics Subject Classification ID: 39B22
Mathematics Subject Classification ID: 03B52
Mathematics Subject Classification ID: 26B40
Mathematics Subject Classification ID: 39B62
zbMATH DE Number: 2113890
zbMATH Keywords: functional equation
zbMATH Keywords: continuous functions
zbMATH Keywords: monotonic functions
zbMATH Keywords: invariant (selfconjugate) operations, invariant domains
zbMATH Keywords: probabilistic metric spaces
zbMATH Keywords: fuzzy implications
reviewed by: János Aczél
MaRDI profile type: Publication
full work available at URL: s00010-004-2737-7
OpenAlex ID: W1576518809
DOI: 10.1007/S00010-004-2737-7
description: scientific article; zbMATH DE number 2113890