Varieties of infinitary universal algebras
type: Item
instance of: scholarly article
title: Varieties of infinitary universal algebras
zbMATH Open document ID: 0559.08002
DOI: 10.1007/BF01236811
author: Jiří Rosický
published in: Algebra Universalis
publication date: 1985-01-01T00:00:00Z
review text: Let \(\tau\) be a type with a proper class of operation symbols. We say that an equational theory E for algebras of type \(\tau\) is legitimate if algebras of type \(\tau\) satisfying E form a class. A subclass V of algebras satisfying a legitimate theory E is a variety if it is closed under products, subalgebras and homomorphic images. The author shows that a universe is weakly compact if and only if each variety can be described by means of equalities between implicit operations (i.e. a variety is a class of algebras of type \(\tau\) satisfying the theory E and some equations between implicit operations). Thus if a universe is weakly compact then we can generalize Reiterman's theorem for finite algebras to algebras of type \(\tau\) satisfying theory E.
reviewed by: Václav Koubek
Mathematics Subject Classification ID: 08A65
Mathematics Subject Classification ID: 03C05
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 08C10
Mathematics Subject Classification ID: 08B99
zbMATH DE Number: 3891439
zbMATH Keywords: weakly compact universe
zbMATH Keywords: proper class of operation symbols
zbMATH Keywords: equational theory
zbMATH Keywords: legitimate theory
zbMATH Keywords: implicit operations
MaRDI profile type: Publication
cites work: Structure-semantics relation for non-varietal equational theories
cites work: The Birkhoff theorem for finite algebras
full work available at URL: https://doi.org/10.1007/bf01236811
OpenAlex ID: W2150186510
description: scientific article; zbMATH DE number 3891439