The translation equation on certain \(n\)-groups
type: Item
instance of: scholarly article
title: The translation equation on certain \(n\)-groups
zbMATH Open document ID: 0798.39008
DOI: 10.1007/BF01838136
author: Andrzej Mach
published in: Aequationes Mathematicae
publication date: 1994-10-30T00:00:00Z
full work available at URL: https://eudml.org/doc/137564
review text: Let \(\Gamma\) be an arbitrary set and \((G, \circ)\) be a group. Define an \(n\)-group operation in \(G\) as follows: \([x_ 1, x_ 2 \dots x_ n] = x_ 1 \cdot x_ 2 \cdot \cdots \cdot x_ n\). Let \(n \geq 3\) be a natural number and \(\varphi\) be a map from \(\Gamma \times G\) to \(\Gamma\). The author considers the translation equation \[ \varphi \Bigl( \varphi \biggl( \varphi \bigl( \dots \varphi (\alpha,x_ 1), x_ 2 \bigr), \dots,x_{n-1}\biggr), x_ n \Bigr) = \varphi (\alpha, x_ 1 \cdot x_ 2 \cdot \cdots \cdot x_ n), \] where \(\alpha \in \Gamma\) and \(x_ 1, x_ 2, \dots, x_ n \in G\), and shows, that in order to characterize all solutions of the above said translation equation it is necessary and sufficient to describe all solutions of the translation equation \(F(F(\alpha,x), y) = F(\alpha,x\cdot y)\) where \(F:\Gamma \times G \to \Gamma\) is a map fulfilling certain conditions.
Mathematics Subject Classification ID: 39B52
zbMATH DE Number: 536986
zbMATH Keywords: group
zbMATH Keywords: translation equation
reviewed by: Harkrishan L. Vasudeva
MaRDI profile type: Publication
cites work: Q5729340
cites work: On solutions of the translation equation commuting with some function
cites work: On the general theory of m-groups
cites work: Structure de l'automate plein, reduit et inversible
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description: scientific article; zbMATH DE number 536986