A group-theoretic approach to covering systems
type: Item
instance of: scholarly article
zbMATH Open document ID: 1362.05138
publication date: 2016-04-13T00:00:00Z
Mathematics Subject Classification ID: 05E18
Mathematics Subject Classification ID: 11B25
Mathematics Subject Classification ID: 11A07
zbMATH DE Number: 6568279
zbMATH Keywords: covering system
zbMATH Keywords: covering number
zbMATH Keywords: group action
zbMATH Keywords: congruence
zbMATH Keywords: holomorph
MaRDI profile type: Publication
author: Lenny Jones
author: Daniel White
title: A group-theoretic approach to covering systems
published in: Algebra and Discrete Mathematics
review text: A covering (system) of the integers is a finite system of congruences \(x\equiv r_i\pmod{n_i}\) for \(1\leq i\leq t\), such that every integer satisfies at least one of these congruences. The authors demonstrate a possibility how to impose an algebraic structure on the set of all coverings with a fixed set of moduli to describe relationships among the elements of the set \(\Gamma_M\) of all coverings having moduli in a finite set of moduli \(M\). In the center of their algebraic approach is the group action of the holomorph \({\mathcal{G}}=\Aut({\mathbb Z}_L)\ltimes{\mathbb Z}_L)\), where \({\mathbb Z}_L\) is the additive group of integer modulo \(L=\text{lcm}(M)\). For instance, there is a natural (left) action of \({\mathcal G}\) on \(\Gamma_M\) (Theorem 4.3). This action is used to enumerate and categorize the elements of \(\Gamma_M\).
reviewed by: Štefan Porubský
description: scientific article; zbMATH DE number 6568279