Clones, order varieties, near unanimity functions and holes
type: Item
instance of: scholarly article
title: Clones, order varieties, near unanimity functions and holes
zbMATH Open document ID: 0734.06001
DOI: 10.1007/BF00418652
author: Ivan Rival
author: Robert W. Quackenbush
author: Ivo G. Rosenberg
published in: Order
publication date: 1990-01-01T00:00:00Z
review text: For an integer \(n\geq 3\) a function m: \(P^ n\to P\) is called a near unanimity function if \(m(x,y,y,...,y)=m(y,x,y,...,y)=...=m(y,y,...,y,x)=y\) for x,y\(\in P\). An ordered set \(P=\{a_ 1,a_ 2,...,a_ n\}\) is called a fence if the ordering in P is given by \(a_ 1<a_ 2>a_ 3<..\). or dually, with no other comparabilities. Theorem 1. A finite ordered set has an order-preserving majority function (i.e. an order-preserving near unanimity function with \(n=3)\) if and only if it is contained in the order variety generated by all fences. The set Cl(P) of all order-preserving functions on an ordered set P forms a clone. The authors put forward the conjecture that a finite bounded ordered set P has an n-ary order-preserving near unanimity function for some \(n\geq 3\) if and only if Cl(P) is finitely generated.
Mathematics Subject Classification ID: 06A06
Mathematics Subject Classification ID: 08A40
zbMATH DE Number: 4214074
zbMATH Keywords: near unanimity function
zbMATH Keywords: fence
zbMATH Keywords: majority function
zbMATH Keywords: order variety
zbMATH Keywords: order- preserving functions
zbMATH Keywords: clone
reviewed by: V. N. Saliĭ
MaRDI profile type: Publication
cites work: Algebraic properties of crowns and fences
cites work: Bestimmung der Ordnung Maximaler Klassen von Funktionen der k‐Wertigen Logik
cites work: Q3730636
cites work: Holes in ordered sets
cites work: The strong selection property and ordered sets of finite length
cites work: An application of the Helly property to the partially ordered sets
cites work: A maximal clone of monotone operations which is not finitely generated
full work available at URL: https://doi.org/10.1007/bf00418652
OpenAlex ID: W1982129048
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description: scientific article; zbMATH DE number 4214074