Subreducts and Subvarieties of PBZ*--lattices
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.
- Pub Date:
- April 2019
- Mathematics - Rings and Algebras;
- 22 pages