The median stabilization degree of a median algebra
type: Item
instance of: scholarly article
title: The median stabilization degree of a median algebra
zbMATH Open document ID: 0916.06010
DOI: 10.1023/A:1018689708341
author: Hans-Jürgen Bandelt
author: Marcel van de Vel
published in: Journal of Algebraic Combinatorics
publication date: 1999-04-13T00:00:00Z
review text: A median algebra consists of a set \(M\) and a median operator on \(M\), which is a symmetric function \(m: M^3\to M\) such that \(m(a,a,b)= a\) and \(m(m(a,b,c),d,c)= m(a, m(b,c,d), c)\). The median stabilization degree (msd, for short) of a median algebra measures the largest possible number of steps needed to generate a subalgebra with an arbitrary set of generators. In this paper, the authors determine the value of the msd of a graphic \(n\)-cube \(Q_n\) and derive an estimation of the msd for the natural median operator of \(\mathbb{R}^n\) which is sharp up to one or two units. Interestingly, the msd of \(Q_n\) and \(\mathbb{R}^n\) grows like \(\log_{15}n\). Finally, the authors characterize median algebras and median graphs of \(\text{msd}\leq 1\) in terms of forbidden subspaces.
Mathematics Subject Classification ID: 06D99
Mathematics Subject Classification ID: 08A62
Mathematics Subject Classification ID: 08A30
Mathematics Subject Classification ID: 52A01
zbMATH DE Number: 1275693
zbMATH Keywords: convex structure
zbMATH Keywords: graphic cube
zbMATH Keywords: superextensions
zbMATH Keywords: median algebra
zbMATH Keywords: median operator
zbMATH Keywords: median stabilization degree
zbMATH Keywords: median graphs
zbMATH Keywords: forbidden subspaces
reviewed by: Dimitru Buşneag
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1275693