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- W3195444310 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 $$mathbb {PBZL}^{*}$$ which includes orthomodular lattices with an extended signature (with the two complements coinciding), as well as antiortholattices (whose Brouwer complements are trivial). We establish a lattice isomorphism between the lattice of subvarieties of the variety $$mathbb {SAOL}$$ 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 $$mathbb {PKA}$$ of pseudo–Kleene algebras, which proves that $$mathbb {SAOL}$$ and thus $$mathbb {PBZL}^{*}$$ has uncountably many subvarieties since $$mathbb {PKA}$$ does and also gives us axiomatizations for all subvarieties of $$mathbb {SAOL}$$ from those of the subvarieties of $$mathbb {PKA}$$ ; furthermore, it proves that the variety $$mathbb {PKA}$$ is generated by the class of the bounded involution lattice reducts of the members of $$mathbb {SAOL}$$ and thus of those of any subvariety of $$mathbb {PBZL}^{*}$$ that includes $$mathbb {SAOL}$$ , hence neither of these classes is a variety. We also obtain an infinity of pairwise disjoint infinite ascending chains of subvarieties of $$mathbb {SAOL}$$ ." @default.
- W3195444310 created "2021-08-30" @default.
- W3195444310 creator A5006529371 @default.
- W3195444310 date "2023-08-01" @default.
- W3195444310 modified "2023-10-18" @default.
- W3195444310 title "Subreducts and Subvarieties of PBZ *–lattices*" @default.
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- W3195444310 doi "https://doi.org/10.1007/s10773-023-05413-w" @default.
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