The class of Kleene algebras satisfying an interpolation property and Nelson algebras
type: Item
instance of: scholarly article
title: The class of Kleene algebras satisfying an interpolation property and Nelson algebras
zbMATH Open document ID: 0621.06009
DOI: 10.1007/BF01230621
published in: Algebra Universalis
publication date: 1986-01-01T00:00:00Z
review text: The paper intends to clarify the role of the interpolation property in the theory of Kleene algebras and in the representation theory of Nelson algebras. Firstly it is shown that the Kalman construction, which associates each bounded distributive lattice \({\mathcal L}\) to a Kleene algebra K(\({\mathcal L})\), yields a functor K and that K has a left adjoint L. The adjunction is an equivalence, if \({\mathcal L}\) is restricted to the category of the centered Kleene algebras satisfying the interpolation property. By restricting K to subcategories, various adjunctions are obtained in the paper: adjunctions from the category of Heyting algebras to the one of Nelson algebras, from the one of Boolean algebras to the one of three valued Lukasiewicz algebras, and from the one of Stone algebras to the one of regular \(\alpha\)-De Morgan algebras. The characterizations of injective Nelson algebras, of subdirectly irreducible ones, and of simple ones are given by using the adjunctions. The congruence lattices of Nelson algebras and of \(\alpha\)-De Morgan algebras are also studied.
reviewed by: Hiroshi Yutani
Mathematics Subject Classification ID: 06D30
Mathematics Subject Classification ID: 03G25
Mathematics Subject Classification ID: 08C05
Mathematics Subject Classification ID: 18A40
zbMATH DE Number: 4006314
zbMATH Keywords: interpolation property
zbMATH Keywords: Nelson algebras
zbMATH Keywords: adjunction
zbMATH Keywords: equivalence
zbMATH Keywords: centered Kleene algebras
zbMATH Keywords: category of Heyting algebras
zbMATH Keywords: Boolean algebras
zbMATH Keywords: Lukasiewicz algebras
zbMATH Keywords: Stone algebras
zbMATH Keywords: De Morgan algebras
zbMATH Keywords: injective Nelson algebras
zbMATH Keywords: subdirectly irreducible
zbMATH Keywords: congruence lattices
author: Roberto Cignoli
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4006314