Braids and their monotone clones
type: Item
instance of: scholarly article
title: Braids and their monotone clones
zbMATH Open document ID: 0811.08004
DOI: 10.1007/BF01191536
author: J. B. Nation
author: Péter P. Pálfy
author: Brian A. Davey
author: Ralph McKenzie
published in: Algebra Universalis
publication date: 1995-05-01T00:00:00Z
review text: The authors investigate properties of monotone clones of ordered sets called braids. For an ordered set \(P\), let \(d(a,b)\) denote the minimal number of edges of the up-fence and the down-fence from \(a\) to \(b\). The reach \(r(P)\) is the supremum of \(d(a,b)\) as \(a\) and \(b\) vary over \(P\). An ordered set \(B\) is called a braid if it has at least three elements, is connected, has finite reach and for any \(a\in B\) there exists a unique \(a'\in B\) with \(d(a,a')= r(B)\). So every \(n\)-crown \(C_ n\) with \(2n\) elements is a braid of reach \(r(C_ n)=n\); if \(L\) is an atomless Boolean lattice, then \(B=L \setminus \{0,1\}\) is a braid with \(r(B)=3\). The main result is the following: If \(B\) is a braid with reach \(r(B)>2\), then the only idempotent order preserving functions \(f: B^ n\to B\) are the \(n\) projections. Hence, using Mal'tsev conditions for congruence modularity, no algebra of monotone functions on a finite braid \(B\) with \(r(B)>2\) generates a congruence modular variety. The main result is also generalized to the so-called near-braids.
reviewed by: Ivan Chajda
Mathematics Subject Classification ID: 08A40
Mathematics Subject Classification ID: 06A06
Mathematics Subject Classification ID: 08B10
zbMATH DE Number: 679600
zbMATH Keywords: fence
zbMATH Keywords: crown
zbMATH Keywords: monotone clones
zbMATH Keywords: ordered sets
zbMATH Keywords: braids
zbMATH Keywords: reach
zbMATH Keywords: order preserving functions
zbMATH Keywords: congruence modularity
zbMATH Keywords: algebra of monotone functions
zbMATH Keywords: near-braids
MaRDI profile type: Publication
cites work: Monotone clones and congruence modularity
cites work: Q4692885
cites work: Algebraic properties of crowns and fences
full work available at URL: https://doi.org/10.1007/bf01191536
OpenAlex ID: W2056084621
description: scientific article; zbMATH DE number 679600