Free Algebras in Certain Varieties of Distributive Pseudocomplemented De Morgan Algebras

Abstract

In this paper we characterize the join irreducible elements of the free algebras on n free generators in the subvarieties of the variety VO of pseudocomplemented De Morgan algebras satisfying the identity zz'* = (zz'*)'*. Mathematics Subject Classification: 03G25, 06D30.

Classical Modal De Morgan Algebras

“…A classical -De Morgan algebra is a De Morgan algebra endowed with an interior operator satisfying an additional condition. Moreover, the variety CMD is intimately connected with some well-known varieties of De Morgan algebras with pseudocomplementation, like the variety V 0 introduced by H. P. Sankappanavar [13] (see also [7]), or the variety S of Involutive Stone algebras introduced by R. Cignoli and M. S de Gallego in [5].…”

“…We shall give an equational characterization of the variety of De Morgan-Stone algebras as De Morgan algebras with an additional modal operator Δ. As a consequence we have also obtained a characterization of the variety V 0 studied by [13] and [7].…”

Varieties of Regular Pseudocomplemented de Morgan Algebras

“…Pseudocomplemented de Morgan algebras were first investigated in Romanowska [20] and have been considered by several authors since (see, for example, Denecke [7], Gaitán [8], Guzmán and Squier [9], as well as [21], [22], [23], [24], [28]). Romanowska's characterisation of finite subdirectly irreducible pm-algebras in [20], was extended to a characterisation of all subdirectly irreducible pm-algebras in [21] (for those algebras which are non-regular) and in [23] (for those algebras which are regular).…”

Congruence properties of pseudocomplemented De Morgan algebras

“…In [22], Romanowska initiated the investigation into the variety PCDM of pseudocomplemented De Morgan algebras by characterizing the finite subdirectly irreducidle PCDM-algebras. Thereafter, several authors pursued the study of these algebras, sometimes in a more general context (cf., e.g., [15][16][17][23][24][25][26]). In the last four papers just mentioned, the first author investigated, among other things, some congruence properties.…”