×

On the characterization of a subvariety of semi-De Morgan algebras. (English) Zbl 1131.06002

In this paper, the authors characterize, by a new set of axioms, the largest subvariety of semi-De Morgan algebras with the congruence extension property. The equational class of semi-De Morgan algebras, introduced by Sankappanavar and denoted by SDMA for short, consists of bounded lattices with an additional unary operation and it contains the variety of pseudocomplemented distributive lattices and \(\mathcal{K}_{1,1}\), one of the subvarieties of Ockham algebras which includes De Morgan algebras. D. Hobby developed a duality for SDMA which he used to find the largest subvariety of SDMA (denoted by \(\mathcal{C}\)) with the congruence extension property. This variety contains both \(\mathcal{K}_{1,1}\) and the equational class of demi-pseudocomplemented lattices (a generalization of pseudocomplemented lattices studied by Sankappanavar) and it is characterized by the following inequalities:
(\(\alpha\)) \(a'\vee b'\geq (a\wedge b)'\wedge (a\wedge c)'\wedge (b\wedge c)'\wedge (b\wedge c')'\),
(\(\beta\)) \(a'\vee (a'\wedge b\wedge b')'\geq (a\wedge b)'\).
The authors had determined (in previous papers) the equations defining principal congruences as well as the subdirectly irreducibles of the variety \(\mathcal{C}\) and, in this paper, they find a new inequality
(\(\gamma\)) \((a'\wedge(b\wedge(c\vee c'))')\vee(b'\wedge (a\wedge c)')=(a\wedge b)'\wedge (a\wedge c)'\wedge (b\wedge (c\vee c'))'\)
such that \(\mathcal{C}\) can be characterized (easier) by \(\gamma\) and \(\beta\).

MSC:

06B20 Varieties of lattices
06D15 Pseudocomplemented lattices
06D30 De Morgan algebras, Łukasiewicz algebras (lattice-theoretic aspects)
PDFBibTeX XMLCite