Subreducts and Subvarieties of PBZ*--lattices

TL;DR

This paper explores the structure and subvarieties of PBZ*-lattices, revealing their algebraic properties, isomorphisms, and chains of subvarieties, with implications for quantum logic models.

Contribution

It establishes a lattice isomorphism between subvarieties of antiortholattices with Strong De Morgan property and pseudo-Kleene algebras, and characterizes the subvarieties of PBZ*-lattices.

Findings

Lattice isomorphism between subvarieties of SAOL and PKA.

Existence of infinite ascending chains of subvarieties.

Characterization of subvarieties in distributive PBZ*-lattices.

Abstract

PBZ*-lattices are bounded lattice-ordered structures endowed with two complements, called Kleene and Brouwer; by definition, they are the paraorthomodular Brouwer-Zadeh lattices in which the pairs of elements with their Kleene complements satisfy the Strong De Morgan condition. These algebras arise in the study of Quantum Logics and they form a variety PBZL* which includes orthomodular lattices with an extended signature (with the two complements coinciding), 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. We establish a lattice isomorphism between the lattice of subvarieties of the variety SAOL generated by the antiortholattices with the Strong De Morgan property and the ordinal sum of the three-element chain with the lattice of subvarieties of the variety PKA of pseudo-Kleene algebras, which also gives us axiomatizations for all subvarieties of SAOL from those of the subvarieties of PKA and proves that the variety PKA is generated by the class of the bounded involution lattice reducts of the members of SAOL and thus of those of any subvariety of PBZL* that includes SAOL, hence neither of these classes is a variety. We also obtain an infinity of pairwise disjoint infinite ascending chains of varieties of PBZ*-lattices, out of which one is formed of subvarieties of SAOL and another one from subvarieties of the variety of distributive PBZ*-lattices.