Heyting algebras. Duality theory. Translated from the Russian by A. Evseev
type: Item
instance of: scholarly article
title: Heyting algebras. Duality theory. Translated from the Russian by A. Evseev
zbMATH Open document ID: 1436.06001
DOI: 10.1007/978-3-030-12096-2
author: Leo Esakia
published in: Trends in Logic -- Studia Logica Library
publication date: 2019-02-18T00:00:00Z
review text: The book under review is a translation from Russian of the book originally published in 1985 [Zbl 0601.06009]. The original edition instigated the research in the duality theory which nowadays has turned into a separate branch of logic. Despite the fact that the book had been published in Russia, it was not readily available even to the Russian-speaking logicians. Even though during 35 years very many fundamental results in duality theory have been obtained, the translation of the book provides an access to the basis on which it holds. The book contains numerous very informative editorial notes that also help to align the emerging terminology of 1980s with today's commonly used terminology. There are three chapters in the book. The first chapter is called ``Preliminary notions and necessary facts'', and it targets the readers who are not very familiar with universal algebras, categories, topology, ordered sets, clusters, and Heyting lattices. The second chapter is called ``Heyting algebras and closure theory''. It starts with the definitions of Heyting and closure algebras and some of their properties proved. In particular, the correspondence between categories of Heyting and closure algebras is established. Then, the modal and superintuitionistic (or in different terms, intermediate) logics are introduced and their relations to closure and Heyting algebras as their algebraic models are discussed. As an alternative type of models, the Kripke models are defined. The subset of all open elements of a given closure algebra \((B,C)\) form a a Heyting algebra called a skeleton. The second part of this chapter is dedicated to a study of skeletal filters (open filters) of the closure algebras, and especially the skeletal closure algebras algebras -- closure algebras generaterd by their skeletons. The third chapter introduces the duality theory. A triple \((X,\Omega,R)\) is hybrid if \begin{itemize} \item[(1)] \((X,\Omega)\) is a Stone space; \item[(2)] \((X,R)\) is a quasi-…
reviewed by: Alex Citkin
Mathematics Subject Classification ID: 06-02
Mathematics Subject Classification ID: 06D20
Mathematics Subject Classification ID: 03B45
Mathematics Subject Classification ID: 06D50
Mathematics Subject Classification ID: 06E15
zbMATH DE Number: 7024700
zbMATH Keywords: Heyting algebras
zbMATH Keywords: closure algebras
zbMATH Keywords: superintuitionistic logic
zbMATH Keywords: modal logic
zbMATH Keywords: duality theory
zbMATH Keywords: hybrid models
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-030-12096-2
OpenAlex ID: W4237415506
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description: scientific article; zbMATH DE number 7024700