Partial orders induced by quasilinear clones
type: Item
instance of: scholarly article
zbMATH Open document ID: 1284.08008
author: Erkko Lehtonen
author: Ágnes Szendrei
publication date: 2013-07-10T00:00:00Z
Mathematics Subject Classification ID: 08A40
Mathematics Subject Classification ID: 03B50
Mathematics Subject Classification ID: 06A06
zbMATH DE Number: 6186384
zbMATH Keywords: clone
zbMATH Keywords: minor
zbMATH Keywords: semimodule
zbMATH Keywords: Burle's clone
MaRDI profile type: Publication
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title: Partial orders induced by quasilinear clones
review text: Let \(A\) and \(U\) be sets, let \({\mathcal C}\) be a clone of operations on \(A\), let \(f:\;A^m\to U\), \(g:\;A^n\to U\) be functions. We write \(f\leq_{\mathcal C}g\) if \(f=g(h_1,\dots,h_n)\) for some \(m\)-ary operations \(h_1,\dots,h_n\in{\mathcal C}\). This defines a quasiorder relation on the set \({\mathcal F}(A,U)\) of all finitary functions \(A^n\to U\). After factorizing by the associated equivalence we obtain the partially ordered set \({\mathcal F}(A,U)/\equiv_{\mathcal C}\). As their main result the authors find several types of clones \({\mathcal C}\) for which the following holds: (1) \({\mathcal F}(A,U)/\equiv_{\mathcal C}\) is countable and all its principal ideals are finite; (2) every countable partially ordered set with finite principal ideals embeds into \({\mathcal F}(A,U)/\equiv_{\mathcal C}\).NEWLINENEWLINEFor the entire collection see [Zbl 1234.08003].
reviewed by: Miroslav Ploščica
description: scientific article; zbMATH DE number 6186384