Distributive PBZ $$^{*}$$ -lattices

Abstract

Arising in the study of Quantum Logics, PBZ\(^{*}\)-lattices are the paraorthomodular Brouwer–Zadeh lattices in which the pairs of elements with their Kleene complements satisfy the Strong De Morgan condition with respect to the Brouwer complement. They form a variety \(\mathbb {PBZL}^{*}\) which includes that of orthomodular lattices considered with an extended signature (by endowing them with a Brouwer complement coinciding with their Kleene complement), as well as antiortholattices (whose Brouwer complements are trivial). The former turn out to have directly irreducible lattice reducts and, under distributivity, no nontrivial elements with bounded lattice complements, since the elements with bounded lattice complements coincide to the sharp elements in distributive PBZ\(^{*}\)-lattices. The variety \(\mathbb {DIST}\) of distributive PBZ\(^{*}\)-lattices has an infinite ascending chain of subvarieties, and the variety generated by orthomodular lattices and antiortholattices has an infinity of pairwise disjoint infinite ascending chains of subvarieties.