Small congruence distributive varieties: retracts, injectives, equational compactness and amalgamation
type: Item
instance of: scholarly article
title: Small congruence distributive varieties: retracts, injectives, equational compactness and amalgamation
zbMATH Open document ID: 0880.08005
DOI: 10.1007/BF00150835
published in: Periodica Mathematica Hungarica
publication date: 1998-01-05T00:00:00Z
review text: The authors consider the relationship between absolute retracts, injectives and equationally compact algebras in finitely generated congruence distributive varieties of algebras with 1-element subalgebras. In particular, they prove that the absolute retracts in such a variety are precisely the injectives in the amalgamation class, that every equationally compact reduced power of a finite absolute retract is an absolute retract. It is shown that any elementary amalgamation class is Horn if and only if it is closed under finite direct products. Some applications to lattice varieties are given.
Mathematics Subject Classification ID: 08B10
Mathematics Subject Classification ID: 08A45
zbMATH DE Number: 1099695
zbMATH Keywords: congruence distributive varieties
zbMATH Keywords: absolute retracts
zbMATH Keywords: equationally compact algebras
zbMATH Keywords: amalgamation
zbMATH Keywords: Horn classes
zbMATH Keywords: injectives
zbMATH Keywords: residually small varieties
author: Peter Ouwehand
author: Henry Rose
reviewed by: Boris Vernikov
MaRDI profile type: Publication
cites work: Equational compactness in equational classes of algebras
cites work: Amalgamation classes of some distributive varieties
cites work: Non-axiomatizability of the amalgamation class of modular lattice varieties
cites work: Boolean powers
cites work: Q3934450
cites work: Weak Injectivity and Congruence Extension in Congruence-Distributive Equational Classes
cites work: Injectives in non-distributive equational classes of lattices are trivial
cites work: The Structure of Pseudocomplemented Distributive Lattices. II: Congruence Extension and Amalgamation
cites work: The amalgamation property in equational classes of modular lattices
cites work: Absolute Retracts and Amalgamation in Certain Congruence Distributive Varieties
cites work: Varieties of lattices
cites work: Amalgamation in small varieties of lattices
cites work: Q3342632
cites work: Filtral powers of structures
cites work: Nonmodular lattice varieties
cites work: Residually small varieties
cites work: Equationally compact algebras (I)
cites work: The Amalgamation Property, the Universal-Homogeneous Models, and the Generic Models.
full work available at URL: https://doi.org/10.1007/bf00150835
OpenAlex ID: W1984236085
description: scientific article; zbMATH DE number 1099695