Abstract
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.
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A corresponding result in group theory is that a finite group is cyclic if and only if its subgroup lattice is distributive (Birkhoff 1967).
A projection p on a Hilbert space H is an operator on H which is self-adjoint, \((p(x),y)=(x,p(y))\), and idempotent, \(p^{2}=p\). Projections are in a bijective correspondence with the closed subspaces of H: If p is a projection, its range \(\mathrm{ran}(p)\) is closed, and any closed subspace is the range of a unique projection, defining: \(p\le q\), if \(\mathrm{ran}(p)\subseteq \mathrm{ran}(q)\), and \(p'=1-p\), where p, q are projection operators on H. Then the set L(H) of projections on H is a lattice.
The bijectivity of \('\) is a very strong condition, which is essential to represent the two arrows in many non-commutative logical algebras by one, cf. Yang and Rump (2012).
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Acknowledgements
The work is partially supported by NSFC (Grant 11271040).
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Communicated by Y. Yang.
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Wu, Y., Yang, Y. Notes on quantum logics and involutive bounded posets. Soft Comput 21, 2513–2519 (2017). https://doi.org/10.1007/s00500-017-2579-6
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DOI: https://doi.org/10.1007/s00500-017-2579-6