Priestley duality for bilattices
type: Item
instance of: scholarly article
title: Priestley duality for bilattices
zbMATH Open document ID: 1258.06008
published in: Studia Logica
publication date: 2012-10-01T00:00:00Z
review text: A pre-bilattice is a set equipped with two separate lattice structures. Such algebras have been used as the basis for a denotational semantics for systems of inference that arise in artificial intelligence and knowledge-based logic programming. A pre-bilattice is distributive if all possible distributive laws hold. A bilattice is a pre-bilattice equipped with an additional unary involution (negation) which reverses both induced order relations. Every (distributive) pre-bilattice can be represented as a special direct product of two (distributive) lattices. Every (distributive) bilattice can be represented as a special direct product of a (distributive) lattice with itself. In fact, the category of bounded distributive pre-bilattices is equivalent to the product category of bounded distributive lattices with itself. Similarly, the category of bounded distributive bilattices is equivalent to the category of bounded distributive lattices; see [\textit{B. Mobasher} et al., Algebra Univers. 43, No. 2--3, 109--125 (2000; Zbl 1012.06008)]. It is therefore possible to use Priestley duality for the representation of bounded distributive pre-bilattices and bilattices. In the present paper, the authors give details of this representation. Next, they consider pre-bilattices and bilattices with additional operations and show how to adapt Priestley duality to obtain dualities for bilattices with conflation and Brouwerian bilattices.
reviewed by: Miroslav Ploščica
Mathematics Subject Classification ID: 06D50
Mathematics Subject Classification ID: 06D75
Mathematics Subject Classification ID: 08C20
Mathematics Subject Classification ID: 68Q55
zbMATH DE Number: 6088859
zbMATH Keywords: Priestley duality theory
zbMATH Keywords: bilattices with conflation
zbMATH Keywords: bilattices with implication
zbMATH Keywords: Brouwerian bilattices
author: Achim Jung
author: Umberto Rivieccio
MaRDI profile type: Publication
full work available at URL: s11225-012-9376-0
OpenAlex ID: W1964591144
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DOI: 10.1007/S11225-012-9376-0
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description: scientific article; zbMATH DE number 6088859