Classical modal De Morgan algebras
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 operation. The author characterizes the prime regular filters and in the case of \(\Delta\)-De Morgan algebras he gives a characterization of the maximal regular filters in terms of minimal prime filters. Finally, it is proved that the variety \(\mathcal{TMA}\) has the amalgamation property and the superamalgamation property.
scientific article; zbMATH DE number 6013480
scientific article