A topological duality for some lattice ordered algebraic structures including \(\ell\)-groups
type: Item
instance of: scholarly article
title: A topological duality for some lattice ordered algebraic structures including \(\ell\)-groups
zbMATH Open document ID: 0822.06014
DOI: 10.1007/BF01236503
author: Néstor G. Martínez
published in: Algebra Universalis
publication date: 1995-09-28T00:00:00Z
review text: The author develops a topological duality for implicative lattices. Implicative lattices are distributive lattices with an additional binary operation which is a dual lattice homomorphism in its first coordinate and a lattice homomorphism in the second coordinate. In this duality the implication corresponds to an order-preserving continuous binary operation. The image of this operation does not lie entirely within the prime filters; it may contain trivial filters. This is remedied by using for the dual space of a distributive lattice the Stone space of the lattice obtained by adding a (possibly new) top and bottom, and then keeping the order to be able to recognize the extra maximum and minimum prime filters that are created. This duality for distributive lattices gives a smooth description of the dual space of an unbounded lattice and allows the author to describe the binary operation obtained from the implication as an operation on the dual space. By specializing his duality for implicative lattices, the author obtains topological dualities for \(l\)-groups and for Abelian \(l\)-groups, since these can be viewed as equational subclasses of implicative lattices with a distinguished element. In the case of Abelian \(l\)-groups, the dual spaces are compact ordered Abelian topological semigroups. Even though these are distributive lattices with additional operation(s), the dualities obtained here are not directly related to the ones obtained by \textit{R. Goldblatt} [Ann. Pure Appl. Logic 44, 173-242 (1989; Zbl 0722.08005)] and other authors who have studied dualities and canonical extensions for distributive lattices with operators and other additional operations. The author obtains a binary operation on the dual space rather than just at ternary relation. And as a consequence, the operation obtained does not share the equational properties of the implication that it came from.
Mathematics Subject Classification ID: 06F99
Mathematics Subject Classification ID: 06D99
Mathematics Subject Classification ID: 06F15
Mathematics Subject Classification ID: 54H10
Mathematics Subject Classification ID: 06F20
Mathematics Subject Classification ID: 06F30
zbMATH DE Number: 638594
zbMATH Keywords: lattice-ordered algebraic structures
zbMATH Keywords: topological duality
zbMATH Keywords: implicative lattices
zbMATH Keywords: dual space
zbMATH Keywords: \(l\)-groups
zbMATH Keywords: Abelian \(l\)-groups
zbMATH Keywords: topological semigroups
MaRDI profile type: Publication
cites work: The class of Kleene algebras satisfying an interpolation property and Nelson algebras
cites work: Q5509706
cites work: The Priestley duality for Wajsberg algebras
cites work: Representation of Distributive Lattices by means of ordered Stone Spaces
cites work: Ordered Topological Spaces and the Representation of Distributive Lattices
cites work: Topological representations of distributive lattices and Brouwerian logics
description: scientific article; zbMATH DE number 638594