Quasi-varieties of metric algebras
type: Item
instance of: scholarly article
title: Quasi-varieties of metric algebras
zbMATH Open document ID: 0821.08004
DOI: 10.1007/BF01190761
published in: Algebra Universalis
publication date: 1995-02-22T00:00:00Z
review text: An \(M\)-algebra or a metric algebra is an abstract algebra endowed with a metric which assumes values in \([0, \infty]\). In a natural way the author introduces such notions as \(M\)-isomorphism, \(M\)-subalgebra, reduced \(M\)- product, and he defines operators \(I\), \(S\), \(P^ M_ R\) corresponding to the above operations. An \(M\)-quasivariety is a class of \(M\)-algebras closed under \(I\), \(S\), \(P_ R^ M\). The key notion of this paper is that of equicontinuous satisfaction (it is an analogue of uniform continuity in the usual sense). The author proves that a class of metric algebras is an \(M\)-quasivariety iff it is definable by finite implications and equicontinuously satisfies every finite implication that it satisfies. It is shown that an \(M\)- quasivariety obeys a subdirect representation theorem and admits completion of its members. Also the author finds \(K\)-irreducible members without infinite distances in the class \(K\) of real metric vector spaces.
Mathematics Subject Classification ID: 08A99
Mathematics Subject Classification ID: 08C15
zbMATH DE Number: 724098
zbMATH Keywords: quasivariety
zbMATH Keywords: irreducibility
zbMATH Keywords: metric algebra
zbMATH Keywords: equicontinuous satisfaction
zbMATH Keywords: subdirect representation theorem
zbMATH Keywords: real metric vector spaces
author: Nik Weaver
reviewed by: Alexander Ivanovich Budkin
MaRDI profile type: Publication
cites work: Generalized varieties
full work available at URL: https://doi.org/10.1007/bf01190761
OpenAlex ID: W2021932528
description: scientific article; zbMATH DE number 724098