Butler groups with a single \(\tau\)-adic relation
type: Item
instance of: scholarly article
title: Butler groups with a single \(\tau\)-adic relation
zbMATH Open document ID: 0907.20049
DOI: 10.1007/BF02305159
author: E. M. Kovyazina
published in: Mathematical Notes
publication date: 1999-01-07T00:00:00Z
review text: The following results are proved: Theorem 2. For a torsion-free Abelian group \(B\) of finite rank \(n\) with a \(\tau\)-adic relation, the following conditions are equivalent: 1) \(B\) is a Butler group; 2) the group \(B\) can be defined by a finite system of \(\tau\)-adic relations with rational coefficients; 3) there exists a \(\tau\)-adic relation with piecewise rational coefficients that defines the group \(B\); 4) there exists a \(\tau\)-space of dimension \(n\) that defines the group \(B\) and whose elements are piecewise rational \(\tau\)-adic numbers; 5) any \(\tau\)-space defining \(B\) has the form \(\gamma V\) , where \(\gamma\) is an invertible \(\tau\)-adic number and \(V\) is a space of dimension \(n\) whose elements are piecewise rational numbers. Theorem 3. The system \(\begin{cases} t_1x+s_1y=0, &t_1,s_1\in\mathbb{Q}(\tau_1),\\ \ldots &\ldots\\ t_mx+s_my=0, &t_m,s_m\in\mathbb{Q}(\tau_m),\end{cases}\) defines a Butler group \(B\) of rank 2 with Richman type \(\sigma_1\leq\sigma_2\) where \(\sigma_1=\bigvee_{i\neq j}(\tau_i\wedge\tau_j)\) and \(\sigma_2=\tau_1\vee\dots\vee\tau_m\).
Mathematics Subject Classification ID: 20K15
zbMATH DE Number: 1158971
zbMATH Keywords: Butler groups
zbMATH Keywords: torsion-free Abelian groups of finite rank
zbMATH Keywords: Richman types
zbMATH Keywords: \(\tau\)-adic relations
reviewed by: Grigore Călugăreanu
MaRDI profile type: Publication
cites work: A Class of Torsion-Free Abelian Groups of Finite Rank
cites work: Finite rank torsion free Abelian groups and rings
cites work: Q4015751
full work available at URL: https://doi.org/10.1007/bf02305159
OpenAlex ID: W2061971863
description: scientific article; zbMATH DE number 1158971