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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
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).
References
Beran L (1985) Orthomodular lattices: algebraic approach. Erschienen Dordrecht: Reidel
Birkhoff G (1967) Lattice theory, 3nd edn, vol 25. AMS Colloquium Publications
Birkhoff G, von Neumann J (1936) The logic of quantum mechanics. Ann. Math. 37(4):823–843
Carrega JC (1995) Minimal orthomodular lattices. Proc Int Quantum Struct Assoc Quantum Struct Int J Theor Phys 34(8):1265–1270
Carrega JC, Fort M (1983) Un problme d’exclusion de treillis orthomodulaires. Comptes Rendus de l’Acad Sci Paris 296:480–496
Chiara ML, Dalla GR (2002) Quantum logics. In: Gabbay D, Guenthner F (eds) Handbook of philosophical logic. Kluwer, Dordrecht, pp 129–228
de Vries A (2009) Algebraic hierarchy of logics unifying fuzzy logic and quantum logic. Lecture Notes, pp 1–65
Giuntini R (1991) A semantical investigation on Brouwer–Zadeh logic. Philos Logic 20(4):411–433
Husimi K (1937) Studies on the foundations of quantum mechanics I. Proc Physicomath Soc Jpn 19:766–789
Kalmbach G (1983) Orthomodular lattices. Cambridge University Press, Cambridge
MacNeille HM (1937) Partially ordered sets. Trans AMS 42(3):416–460
Yang Y, Rump W (2012) Pseudo-MV algebras as L-algebras. J Multi-valued Logic Soft Comput 19(5–6):621–632
Acknowledgements
The work is partially supported by NSFC (Grant 11271040).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Communicated by Y. Yang.
Rights and permissions
About this article
Cite this article
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
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00500-017-2579-6