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. (shrink)

We continue in this article the abstract algebraic treatment of quantum sentential logics Wil. The Notions borrowed from the field of Model Theory and Abstract Algebraic Logic - AAL (i.e., consequence relation, variety, logical matrix, deductive filter, reduced product, ultraproduct, ultrapower, Frege relation, Leibniz congruence, Suszko congruence, Leibniz operator) are applied to quantum logics. We also proved several equivalences between state property systems (Jauch-Piron-Aerts line of investigations) and AAL treatment of quantum logics (corollary 18 and 19). We show that there (...) exist the uniquely defined correspondence between state property system and consequence relation defined on quantum logics. We also signalize that a metalogical property - Lindenbaum property does not hold for the set of quantum logics. (shrink)

In the present paper we give syntactical and semantical characterization of the class of algebras defined by P-compatible identities of modular ortholattices. We also describe the lattice of some subvarieties of the variety MOLEx defined by so called externally compatible identities of modular ortholattices.