Varieties in which the Pierce stalks are directly indecomposable
type: Item
instance of: scholarly article
title: Varieties in which the Pierce stalks are directly indecomposable
zbMATH Open document ID: 0868.08003
published in: Journal of Algebra
publication date: 1997-08-18T00:00:00Z
review text: If the sublattice of the factor congruences of the congruences of an algebra \(A\) is a Boolean algebra with the Stone space \(X\) of maximal ideals, then the natural map \(A\to\prod\{A/\bigcup m:m\in X\}\) is an isomorphism between \(A\) and the so-called algebra of global sections of the Pierce sheaf. This construction is possible, for example, for the varieties with distributive congruence lattices or for the variety of rings with unit. In this paper, the author characterizes varieties in which the stalks of the Pierce sheaves are directly indecomposable. A variety of algebras is called a Pierce variety if there exists a term \(u= u(x,y,x_1,x_2,\dots,x_n,y_1,y_2,\dots,y_n)\) such that the following two equalities \(u(x,y,w,\vec 0,\vec 1)=x\) and \(u(x,y,w,\vec 1,\vec 0)= y\) hold for some \(n\)-ary terms \(\vec 0\), \(\vec 1\). The following is proved: the stalks of the Pierce sheaf of every element of a variety \({\mathbf V}\) are directly indecomposable iff \({\mathbf V}\) is a Pierce variety such that every subalgebra of any subdirectly irreducible member is directly indecomposable and every ultraproduct of any family of subdirectly irreducible members is directly indecomposable.
Mathematics Subject Classification ID: 08B05
Mathematics Subject Classification ID: 08A30
Mathematics Subject Classification ID: 08B26
zbMATH DE Number: 930150
zbMATH Keywords: factorable congruence
zbMATH Keywords: subdirect product
zbMATH Keywords: stalks
zbMATH Keywords: Pierce sheaves
zbMATH Keywords: Pierce variety
zbMATH Keywords: ultraproduct
author: Diego J. Vaggione
reviewed by: Ernest Płonka
MaRDI profile type: Publication
full work available at URL: jabr.1996.0268
OpenAlex ID: W2055921182
DOI: 10.1006/JABR.1996.0268
description: scientific article; zbMATH DE number 930150