Coproducts of distributive lattice-based algebras.
type: Item
instance of: scholarly article
title: Coproducts of distributive lattice-based algebras.
zbMATH Open document ID: 1312.06009
author: Leonardo Manuel Cabrer
author: Hilary A. Priestley
published in: Algebra Universalis
publication date: 2014-11-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/1308.4650
review text: The study of coproducts in different varieties of algebras can be found in mathematics' literature for a long time, the characterization of coproducts often being motivated by the interest in amalgation properties, determination of free objects, axiomatizations or colimits. The main aim of this paper is to present a uniform theory, based on duality, of coproducts in classes of algebras that admit a bounded distributive lattice reduct. Let \(\mathcal D\) denote the variety of bounded distributive lattices. A class \(\mathcal A\) of algebras with language \(\mathcal L\) is a \(\mathcal D\)-based class if the algebras of \(\mathcal A\) have reduct in \(\mathcal D\). In this paper the authors study coproducts in \(\mathcal D\)-based categories of algebras by relating them to coproducts in \(\mathcal D\) (it is well known the fact that Priestley duality for (bounded) distributive lattices establishes a dual equivalence between the category \(\mathcal D\) and the category \(\mathcal P\) of Priestley spaces). For every \(\mathcal D\)-based class of algebras, \(\mathcal A\), we have a (forgetful) functor \(U_{\mathcal A}\) from \(\mathcal A\) into \(\mathcal D\) and one of the main problems of this paper is to determine when \(U_{\mathcal A}\) preserves coproducts. They also investigate the possible behaviours of this functor as regards coproducts in \(\mathcal A\) under weaker assumptions. The authors restrict their attention to the situation in which \(\mathcal A\) is a finitely generated quasivariety (quasivarieties ensure the existence of coproducts for each set of algebras in the class). The principal results of this paper are presented in Section 3 and Theorem 3.6 (Coproducts Preservation Theorem) is the most important of them. It establishes equivalent conditions for the forgetful functor \(U_{\mathcal A}\) to preserve coproducts. In the final part of this section the authors give a short description of the free product and present the connections between the propert…
reviewed by: Florentina Chirteş
Mathematics Subject Classification ID: 06D50
Mathematics Subject Classification ID: 08C20
Mathematics Subject Classification ID: 08B25
Mathematics Subject Classification ID: 18A35
Mathematics Subject Classification ID: 08C15
Mathematics Subject Classification ID: 06D30
zbMATH DE Number: 6369493
zbMATH Keywords: coproducts
zbMATH Keywords: distributive lattices
zbMATH Keywords: ordered algebras
zbMATH Keywords: functor preserving coproducts
zbMATH Keywords: natural dualities
zbMATH Keywords: piggyback duality
zbMATH Keywords: distributive lattice reducts
MaRDI profile type: Publication
OpenAlex ID: W1984218673
arXiv ID: 1308.4650
cites work: Q4083730
cites work: Locally finite varieties.
cites work: Locally finite varieties of Heyting algebras
cites work: Lukasiewicz-Moisil algebras
cites work: Q3934450
cites work: Gödel algebras: interactive dualities and their applications.
cites work: Coproducts in the categories of Kleene and three-valued Lukasiewicz algebras
cites work: Quantifiers on distributive lattices
cites work: Free \(Q\)-distributive lattices
cites work: Algebraic foundations of many-valued reasoning
cites work: Q4220406
cites work: Coproducts of De Morgan algebras
cites work: Q4187871
cites work: Computing coproducts of finitely presented Gödel algebras
cites work: Dualities for Equational Classes of Brouwerian Algebras and Heyting Algebras
cites work: Q3033816
cites work: Partition-induced natural dualities for varieties of pseudo-complemented distributive lattices
cites work: Optimal natural dualities for varieties of Heyting algebras
cites work: Q4530231
cites work: Piggyback-Dualitäten
cites work: Representation of Lukasiewicz algebras by means of ordered Stone spaces
cites work: Q5416148
cites work: Q4220413
cites work: Q4215784
cites work: Q4123496
cites work: Q3607805
cites work: Advanced Łukasiewicz calculus and MV-algebras
cites work: Natural dualities for varieties of MV-algebras. I
cites work: Varieties of MV-algebras
cites work: Q4273437
cites work: Natural dualities for varieties of \(n\)-valued Łukasiewicz algebras
cites work: Free products of bounded distributive lattices
DOI: 10.1007/S00012-014-0302-3
description: scientific article; zbMATH DE number 6369493