Generating primitive positive clones
type: Item
instance of: scholarly article
title: Generating primitive positive clones
zbMATH Open document ID: 1013.08003
author: John W. Snow
published in: Algebra Universalis
publication date: 2003-04-23T00:00:00Z
review text: A primitive positive formula is a first-order formula of the form \(\exists \;\wedge \) (atomic). A clone \(C\) on a set \(A\) is a primitive positive clone if every operation on \(A\) defined from operations in \(C\) using a primitive positive formula is again in \(C\). \textit{S. Burris} and \textit{R. Willard} [Proc. Am. Math. Soc. 101, 427-430 (1987; Zbl 0656.08002)] proved that there are finitely many primitive positive clones on any finite set. They conjectured that every such primitive positive clone is actually generated from its members of rank \(|A|\) if \(A\) has more than two elements. Suppose that for a finite algebra \textbf{A}, PPC\# (\textbf{A}) is the smallest number \(n\) for which PPC(Clo\textbf{A}), the primitive positive clone generated by Clo\textbf{A}, is equal to PPC(Clo\(_n\)\textbf{A}). In the paper the author discusses the problem how large PPC\# (\textbf{A}) can be when special conditions are assumed for the finite algebra \textbf{A}. It is shown that PPC\# (\textbf{A})\(\leq |A|+2\) if \textbf{A} generates a congruence permutable variety and every subalgebra of \textbf{A} is the product of a congruence neutral algebra and an Abelian algebra. PPC\# (\textbf{A})\(\leq |A|\) holds when the variety generated by \textbf{A} is congruence distributive, Abelian or decidable. Furthermore, an example is given for which PPC\# (\textbf{A})\(\geq (|A|-1)^2.\)
reviewed by: Radomír Halaš
Mathematics Subject Classification ID: 08A40
Mathematics Subject Classification ID: 08A05
Mathematics Subject Classification ID: 08B10
zbMATH DE Number: 1899959
zbMATH Keywords: primitive positive clone
zbMATH Keywords: congruence distributive variety
zbMATH Keywords: Abelian variety
zbMATH Keywords: decidable variety
zbMATH Keywords: congruence permutable variety
zbMATH Keywords: congruence neutral algebra
MaRDI profile type: Publication
full work available at URL: s000120050179
OpenAlex ID: W1990731217
DOI: 10.1007/S000120050179
Recommended article: Q3539582
Recommended article: Finitely Many Primitive Positive Clones
Recommended article: Q3724393
Recommended article: On Boolean primitive positive clones
Recommended article: Q4700368
Recommended article: Q5187334
Recommended article: Q4308342
Recommended article: Q3322162
Recommended article: Decidability of Definability
Recommended article: Algebras, Lattices, Varieties
description: scientific article; zbMATH DE number 1899959