Natural dualities in partnership
type: Item
instance of: scholarly article
title: Natural dualities in partnership
zbMATH Open document ID: 1284.08021
author: Brian A. Davey
author: Miroslav Haviar
author: Hilary A. Priestley
published in: Applied Categorical Structures
publication date: 2013-01-02T00:00:00Z
review text: Priestley duality for the variety \(\mathcal{D}\) of bounded distributive lattices sets up a dual equivalence between \(\mathcal{D}\) and the category of Priestley spaces. Banaschewski's duality sets up a dual equivalence between the category of Boolean topological distributive lattices and the category of ordered sets. These partner dualities are set up by hom-functors into a pair of structures: a structure (with no topology) and a topological structure acting as its \textit{alter ego}. Thus the two dualities are related to one another by topology swapping: each can be obtained from its partner by removing the topology from the \textit{alter ego} and applying it to the untopologised structure. In their paper the authors demonstrate that the above examples illustrate a very general procedure that takes a duality and swaps the topology from the structure side to the algebra side to obtain a new partner duality paired with the original one. The main theorem of this paper validates topology swapping as a uniform way to create new dual adjunctions and dual equivalences. The theorem is illustrated via a variety of natural dualities, some classic and some less familiar.
Mathematics Subject Classification ID: 08C20
Mathematics Subject Classification ID: 06B23
Mathematics Subject Classification ID: 06B30
Mathematics Subject Classification ID: 06D50
zbMATH DE Number: 6121298
zbMATH Keywords: natural duality
zbMATH Keywords: natural extension
zbMATH Keywords: canonical extension
zbMATH Keywords: topology swapping
reviewed by: Dimitru Buşneag
MaRDI profile type: Publication
full work available at URL: s10485-011-9253-4
OpenAlex ID: W1977205581
cites work: Q4173540
cites work: Q4308745
cites work: Q4220406
cites work: Standard topological algebras: syntactic and principal congruences and profiniteness
cites work: Q4679049
cites work: The axiomatizability of topological prevarieties
cites work: Binary homomorphisms and natural dualities
cites work: Dualities for Equational Classes of Brouwerian Algebras and Heyting Algebras
cites work: Dualities for Stone algebras, double Stone algebras, and relative Stone algebras
cites work: Q5446632
cites work: Natural extensions and profinite completions of algebras
cites work: Using coloured ordered sets to study finite-level full dualities
cites work: Boolean topological distributive lattices and canonical extensions
cites work: Full does not imply strong, does it?
cites work: Natural dualities for semilattice-based algebras
cites work: The lattice of alter egos
cites work: Q3033816
cites work: A topological approach to canonical extensions in finitely generated varieties of lattice-based algebras
cites work: Functor category dualities for varieties of Heyting algebras
cites work: Dual categories for endodualisable Heyting algebras: optimization and axiomatization
cites work: Natural dualities for varieties of BL-algebras
cites work: Q4313857
cites work: Distributive Ockham algebras: free algebras and injectivity
cites work: A note on profinite completions and canonical extensions
cites work: On profinite completions and canonical extensions
cites work: Q5472326
cites work: Finitely generated free modular ortholattices. II
cites work: A generalization of the duality compactness theorem
cites work: An algebra that is dualizable but not fully dualizable
cites work: Natural dualities for three classes of relational structures
cites work: Natural dualities for varieties of MV-algebras. I
cites work: Theorems on Compact Totally Disconnected Semigroups and Lattices
cites work: Dualisability. Unary algebras and beyond
cites work: Representation of Distributive Lattices by means of ordered Stone Spaces
cites work: Q4273437
cites work: Q4380501
cites work: Q4208407
cites work: RESIDUAL PROPERTIES OF SIMPLE GRAPHS
cites work: Residual properties of pre-bipartite digraphs
cites work: Q5392460
DOI: 10.1007/S10485-011-9253-4
description: scientific article; zbMATH DE number 6121298