A Note on Direct Products, Subreducts and Subvarieties of PBZ*--lattices
Claudia Mureșan · arXiv (Cornell University) · 2019
PBZ*--lattices are bounded lattice--ordered structures arising in the study of quantum logics, which include orthomodular lattices, as well as antiortholattices. Antiortholattices turn out not only to be directly irreducible, but also to have directly irreducible lattice reducts. Their presence in varieties of PBZ*--lattices determines the lengths of the subposets of dense elements of the members of those varieties. The variety they generate includes two disjoint infinite ascending chains of subvarieties, and the lattice of subvarieties of the variety of pseudo--Kleene algebras can be embedded as a poset in the lattice of subvarieties of its subvariety formed of its members that satisfy the Strong De Morgan condition. We obtain axiomatizations for all members of a complete sublattice of the lattice of subvarieties of this latter variety axiomatized by the Strong De Morgan identity with respect to the variety generated by antiortholattices.