Notes on quantum logics and involutive bounded posets
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In this paper, we study the exclusion problem concerning the classes of involutive bounded lattices, logics, and quantum logics (i.e., orthomodular lattices). We also obtain that a logic is a quantum logic if and only if it is a paraconsistent logic. Moreover, we give some considerations on an open question to find sufficient conditions for the existence of an orthomodular orthocomplementation on lattices. Furthermore, we revisit the Dedekind–MacNeille completion of involutive bounded posets and correct a widely cited error in quantum logics.
KeywordsInvolutive bounded poset Ortholattice Orthomodular lattice (quantum logic) Dedekind–MacNeille completion
The work is partially supported by NSFC (Grant 11271040).
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Conflict of interest
The authors declare that they have no conflict of interest.
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