Orthomodular lattices that are \(Z_2\)-rich
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We study the orthomodular lattices (OMLs) that have an abundance of \(Z_2\)-valued states. We call these OMLs \(Z_2\)-rich. The motivation for the investigation comes from a natural algebraic curiosity that reflects the state of the (orthomodular) art, the consideration also has a certain bearing on the foundation of quantum theories (OMLs are often identified with “quantum logics”) and mathematical logic (\(Z_2\)-states are fundamental in mathematical logic). Before we launch on the subject proper, we observe, for a potential application elsewhere, that there can be a more economic introduction of \(Z_2\)-richness - the \(Z_2\)-richness in the orthocomplemented setup is sufficient to imply orthomodularity. In the further part we review basic examples of OMLs that are \(Z_2\)-rich and that are not. Then we show, as a main result, that the \(Z_2\)-rich OMLs form a large and algebraicly “friendly” class—they form a variety. In the appendix we note that the OMLs that allow for a natural introduction of a symmetric difference provide a source of another type of examples of \(Z_2\)-rich OMLs. We also formulate open questions related to the matter studied.
KeywordsOrthocomplemented poset Quantum logic Projection logic Group-valued state Symmetric difference Boolean algebra Variety of algebras
Mathematics Subject Classification06A11 03G12 28E99 81P10 08B