Polynomial clones on groups of order \(pq\)
type: Item
instance of: scholarly article
title: Polynomial clones on groups of order \(pq\)
zbMATH Open document ID: 1121.08004
author: Erhard Aichinger
author: Peter Mayr
published in: Acta Mathematica Hungarica
publication date: 2007-05-08T00:00:00Z
review text: \textit{P. M. Idziak} [Int. J. Algebra Comput. 9, No. 2, 213--226 (1999; Zbl 1023.08003)] had conjectured that, for any square-free \(n\), the cyclic group of order \(n\) has only finitely many polynomially inequivalent expansions. For \(n = pq\) where \(p\) and \(q\) are distinct primes, this conjecture is confirmed by the following results of the present paper: There are only finitely many constantive clones on a \(pq\)-element set that have some group operation among their binary operations, which follows from the fact that each such clone is uniquely determined by congruences and commutator relations. In particular, there are exactly 17 polynomially inequivalent expansions of the cyclic group of order \(pq\).
Mathematics Subject Classification ID: 08A40
Mathematics Subject Classification ID: 16Y30
zbMATH DE Number: 5150268
zbMATH Keywords: Mal'cev clones
zbMATH Keywords: polynomial functions
zbMATH Keywords: polynomial completeness
zbMATH Keywords: commutator theory
reviewed by: Manfred Armbrust
describes a project that uses: SONATA
MaRDI profile type: Publication
full work available at URL: s10474-006-0530-x
OpenAlex ID: W1964770240
cites work: Q4023361
cites work: Hagemann's and Herrmann's characterization of strictly affine complete algebras
cites work: On near-ring idempotents and polynomials on direct products of \(\Omega\)-groups
cites work: ON THE MAXIMAL IDEALS OF NON-ZERO-SYMMETRIC NEAR-RINGS AND OF COMPOSITION ALGEBRAS OF POLYNOMIAL FUNCTIONS ON Ω-GROUPS
cites work: A BOUND ON THE NUMBER OF UNARY POLYNOMIAL FUNCTIONS AND A DECIDABILITY RESULT FOR NEAR-RINGS
cites work: Polynomial interpolation in expanded groups.
cites work: Completeness for concrete near-rings.
cites work: Polynomial Clones Containing the Mal'tsev Operation of the Groups {\open Z}_{p^{2}} and {\open Z}_p \times {\open Z}_p
cites work: Counting Mal'tsev clones on small sets
cites work: Q3775604
cites work: Q3955414
cites work: The structure of finite algebras
cites work: CLONES CONTAINING MAL'TSEV OPERATIONS
cites work: Polynomially rich algebras
cites work: Three remarks on the modular commutator
cites work: Q3671335
DOI: 10.1007/S10474-006-0530-X
description: scientific article; zbMATH DE number 5150268