Polyadic analogs of the Cayley and Birkhoff theorems
type: Item
instance of: scholarly article
title: Polyadic analogs of the Cayley and Birkhoff theorems
zbMATH Open document ID: 1028.20043
published in: Russian Mathematics
publication date: 2002-06-09T00:00:00Z
review text: Let \(A\) be an \(n\)-ary group with \(n\)-ary operation \(f\). On the set \(A^{n-1}\) define an \(n\)-ary operation \(g\) analogous with the \(n\)-ary operation, defined by Post for the \(n\)-ary transformations, by \[ \begin{multlined} g((a_1',\dots,a_{n-1}'),(a_1'',\dots,a_{n-1}''),\dots,(a_1^{(n)},\dots,a_{n-1}^{(n)}))=\\ =(f(a_1',a_2'',\dots,a_{n-1}^{(n)},a_1^{(n)}),f(a_2',\dots,a_{n-1}^{(n-2)},a_1^{(n-1)},a_2^{(n)}),\dots,f(a_{n-1}',a_1'',\dots,a_{n-1}^{(n)})).\end{multlined} \] The Cartesian product \(A^{n-1}\) endowed with the \(n\)-ary operation \(g\) is an \(n\)-ary group. The author establishes some results analogous with the Cayley and Birkhoff theorems. The main result is the following: For each \(n\)-ary group \((A,f)\) there exists an isomorphism of \((A^{n-1},g)\) with the \(n\)-ary group of all \(n\)-ary automorphisms of some sequence of universal algebras.
Mathematics Subject Classification ID: 20N15
zbMATH DE Number: 1752618
zbMATH Keywords: \(n\)-ary groups
zbMATH Keywords: Cayley theorem
zbMATH Keywords: Birkhoff theorem
zbMATH Keywords: \(n\)-ary automorphisms
author: A. M. Gal'mak
reviewed by: Ilie Corovei
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1752618