Classical modal De Morgan algebras
type: Item
instance of: scholarly article
title: Classical modal De Morgan algebras
zbMATH Open document ID: 1245.03109
published in: Studia Logica
publication date: 2012-03-09T00:00:00Z
review text: The author defines and studies the variety \(\mathcal{ CDM}_\square\) of classical modal De Morgan algebras (classical \(\square\)-De Morgan algebras). This variety is an interesting generalization of the variety \(\mathcal{TMA}\) of tetravalent modal algebras, introduced by A. Monteiro and mainly studied by \textit{I. Loureiro} [``Prime spectrum of a tetravalent modal algebra'', Notre Dame J. Formal Logic 24, 389--394 (1983; Zbl 0524.03053)] and \textit{J. M. Font} and \textit{M. Rius} [``An abstract algebraic logic approach to tetravalent modal logics'', J. Symb. Log. 65, No. 2, 481--518 (2000; Zbl 1013.03075)]. A cassical \(\square\)-De Morgan algebra is a De Morgan algebra endowed with an interior operator \(\square\) satisfying an additional condition. The author shows that the variety \(\mathcal{ CDM}_\square\) is intimately connected with some well-known varieties of De Morgan algebras with pseudocomplementation, like the variety \(\nu_0\) introduced by \textit{H. P. Sankappanavar} [``Principal congruences of pseudocomplemented De Morgan algebras'', Z. Math. Logik Grundlagen Math. 33, 3--11 (1987; Zbl 0624.06016)] or the variety \(\mathbf{S} \) of involutive Stone algebras introduced by \textit{R. Cignoli} and \textit{M. S. De Gallego} [``Dualities for some De Morgan algebras with operators and Łukasiewicz algebras'', J. Aust. Math. Soc., Ser. A 34, 377--393 (1983; Zbl 0523.06019)]. He gives an equational characterization of the variety of De Morgan-Stone algebras as De Morgan algebras with an additional modal operator \(\Delta\). This characterization is similar to the definition of the involutive Stone algebras. As a consequence of this characterization he obtains a new equational definition of the variety \(\nu_0\). Then he studies the representation of classical \(\square\)-De Morgan algebras and the class of filters that are closed under \(\square\), called regular filters. These filters can be characterized as lattice filters closed under an implication…
reviewed by: Florentina Chirteş
Mathematics Subject Classification ID: 03G25
Mathematics Subject Classification ID: 06D30
zbMATH DE Number: 6013480
zbMATH Keywords: modal operator
zbMATH Keywords: De Morgan algebras
zbMATH Keywords: amalgamation property
author: Sergio A. Celani
MaRDI profile type: Publication
full work available at URL: s11225-011-9328-0
OpenAlex ID: W1978382207
cites work: Simple and subdirectly irreducibles bounded distributive lattices with unary operators
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cites work: The algebra of topology
cites work: Principal Congruences of Pseudocomplemented Demorgan Algebras
DOI: 10.1007/S11225-011-9328-0
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description: scientific article; zbMATH DE number 6013480