Generalizations of Boolean products for lattice-ordered algebras
type: Item
instance of: scholarly article
title: Generalizations of Boolean products for lattice-ordered algebras
zbMATH Open document ID: 1181.06005
published in: Annals of Pure and Applied Logic
publication date: 2009-11-04T00:00:00Z
review text: Topological dualities have been very effective tools for various classes of algebras, such as Boolean algebras with Boolean spaces as duals, distributive lattices with Priestley spaces as duals, and Heyting algebras with Esakia spaces as duals. Boolean spaces have also been applied to the representation of algebras by Boolean powers and (weak) Boolean products, where the latter are also known as algebras of global sections of sheaves of algebras over Boolean spaces. In this paper, generalizing both Boolean products and poset sums of residuated lattices, the concepts of poset product, Priestley product and Esakia product of algebras are defined and used to prove decomposition theorems for various ordered algebras. In particular, the author shows that \(\text{FL}_{\omega}\)-algebras decompose as a poset product over any finite set of join-irreducible strongly central elements, and that bounded \(n\)-potent GBL-algebras are represented as Esakia products of simple \(n\)-potent MV-algebras.
reviewed by: Dana Piciu
Mathematics Subject Classification ID: 06F05
Mathematics Subject Classification ID: 06D35
Mathematics Subject Classification ID: 03G25
zbMATH DE Number: 5624911
zbMATH Keywords: residuated lattices
zbMATH Keywords: generalized BL-algebras
zbMATH Keywords: basic logic
zbMATH Keywords: generalized MV-algebras
zbMATH Keywords: posets
author: Peter Jipsen
MaRDI profile type: Publication
full work available at URL: j.apal.2009.05.005
OpenAlex ID: W2076261541
cites work: Q5331549
cites work: Q3934450
cites work: Sheaf Constructions and Their Elementary Properties
cites work: Algebraic foundations of many-valued reasoning
cites work: Boolean products of lattices
cites work: Q4530231
cites work: Boolean products of BL-algebras
cites work: Residuated lattices. An algebraic glimpse at substructural logics
cites work: On the structure of generalized BL-algebras
cites work: The Blok-Ferreirim theorem for normal GBL-algebras and its application
cites work: Embedding theorems for classes of GBL-algebras
cites work: Q5693612
cites work: Q4856909
DOI: 10.1016/J.APAL.2009.05.005
description: scientific article; zbMATH DE number 5624911