Heyting algebras. Duality theory. Translated from the Russian by A. Evseev
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-order; \item[(3)] the relation \(R\) is continuous. \end{itemize} Theorem 3.1.2 provides different criteria for a triple \((X,\Omega,R)\), where \((X,\Omega)\) is a Stone space and \((X,R)\) is a quasi-order, to be a hybrid. A hybrid is strict if its quasi-order is a partial order. The remaining part of the section studies the properties of the strict hybrids. In particular, the duality theorem which establishes the equivalence of the categories of closure algebras and hybrids is proven. In a similar way, the category of Heyting algebras is related to the category of strict hybrids; the category of Heyting algebras is equivalent to the category of strict hybrids, and the category of skeletal closure algebras is equivalent to the category of strict hybrids. The last section is dedicated to the study of the Grzegorczyk algebras, that is, the closure algebras admitting the identity \(\Box(\Box( x \to \Box x) \to x ) \approx x\). These algebras are models for the Grzegorczyk logic \(\mathbf{Grz}\) -- the strongest modal companion of the intuitionistic logic. It is shown that a closure algebra \((B,\mathbf{C})\) is a Ggzegorczyk algebra if and only if there is no \(a \in B\) and natural numbers \(n < m\) such that \(\mathbf{C}^na =\mathbf{C}^ma\). Also, it is proven that the variety \(\mathsf{GA}\) of all Grzegorczyk algebras is generated by its finite members and therefore, the variety \(\mathsf{GA}\) is generated by its skeletal members, because classes of finite Grzegorczyk and skeletal algebras coincide. In the Appendices, the author provides an overview of some topics and results that were intended to be published in the second volume of the monograph, which, unfortunately, never happened.
scientific article; zbMATH DE number 7024700