Classical Modal De Morgan Algebras

Abstract

In this note we introduce the variety CDM of classical modal De Morgan algebras as a generalization of the variety T MA of Tetravalent Modal algebras studied in [11]. We show that the variety V 0 defined by H. P. Sankappanavar in [13], and the variety S of Involutive Stone algebras introduced by R. Cignoli and M. S de Gallego in [5], are examples of classical modal De Morgan algebras. We give a representation theory, and we study the regular filters, i.e., lattice filters closed under an implication operation. Finally we prove that the variety T MA has the Amalgamation Property and the Superamalgamation Property.

Discrete duality for 3-valued Łukasiewicz–Moisil algebras

“…The present paper provides a basis for further work on discrete duality for De Morgan algebras with modal operators (see [2,3]).…”

On PBZ*-lattices

“…Bounded distributive lattices expanded both by a De Morgan complementation and a unary operation with Stone-like properties have been the object of rather intensive investigations over the past decades. In particular, Blyth, Fang and Wang [6] have studied, under the label of quasi-Stone De Morgan algebras, bounded distributive lattices with two unary operations that make their appropriate reducts, at the same time, De Morgan algebras and quasi-Stone algebras [37,17,13]. Quasi-Stone De Morgan algebras that are simultaneously Stone algebras and Kleene algebras are known under the name of Kleene-Stone algebras; they have been studied in [25] and, more recently, in the already quoted [6].…”

“…Condition M5, which is of course trivial once our algebras have a Boolean nonmodal reduct, is there to restore the Boolean behaviour of the nonmodal operators, when applied to arguments of the form ♦x. Observe that all classical ♦-De Morgan algebras satisfy the identity M8 [13,Lemma 2.3].…”