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On residuation in paraorthomodular lattices

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Abstract

Paraorthomodular lattices are quantum structures of prominent importance within the framework of the logico-algebraic approach to (unsharp) quantum theory. However, at the present time it is not clear whether the above algebras may be regarded as the algebraic semantic of a logic in its own right. In this paper, we start the investigation of material implications in paraorthomodular lattices by showing that any bounded modular lattice with antitone involution \({\mathbf {A}}\) can be converted into a left-residuated groupoid if it satisfies a strengthened form of regularity. Moreover, the above condition turns out to be also necessary whenever \({\mathbf {A}}\) is distributive.

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References

  • Beran L (1985) Orthomodular lattices: algebraic approach. Riedel, Dordrecht

    Book  Google Scholar 

  • Birkhoff G, Von Neumann J (1936) The logic of quantum mechanics. Ann Math 37(4):823–843

    Article  MathSciNet  Google Scholar 

  • Borzooei RA, Dvurečenskij A, Sharafi AH (2018) Material implications in lattice effect algebras. Inf Sci 433–434:233–240

    Article  MathSciNet  Google Scholar 

  • Botur M, Chajda I, Halaš R (2010) Are basic algebras residuated structures? Soft Comput 14:251–255

    Article  Google Scholar 

  • Chajda I (2016) A note on pseudo-Kleene algebras. Mathematica 55:39–45

    MathSciNet  MATH  Google Scholar 

  • Chajda I, Halaš R, Kühr J (2009) Every effect algebra can be made into a total algebra. Algebra Universalis 61:139–150

    Article  MathSciNet  Google Scholar 

  • Chajda I, Länger H (2018) Weakly orthomodular and dually weakly orthomodular lattices. Order 35:541–555

    Article  MathSciNet  Google Scholar 

  • Chajda I, Länger H (2017) Residuation in orthomodular lattices. Topol Algebra Appl 5:1–5

    MathSciNet  MATH  Google Scholar 

  • Chajda I, Länger H (xxxx) Residuation in modular lattices and posets. Asian Eur J Math, to appear

  • Chajda I, Länger H (2017) Orthomodular lattices can be converted into left-residuated l-groupoids. Miskolc Math Notes 18:685–689

    Article  MathSciNet  Google Scholar 

  • Dalla Chiara ML, Giuntini R, Greechie R (2004) Reasoning in quantum theory-sharp and unsharp quantum logic. Kluwer, Dordrecht

    Book  Google Scholar 

  • Font JM (1997) Belnap’s four-valued logic and De Morgan lattices. Logic J IGPL 5:1–29

    Article  MathSciNet  Google Scholar 

  • Foulis DJ, Bennett MK (1994) Effect algebras and unsharp quantum logic. Found Phys 24:1331–1352

    Article  MathSciNet  Google Scholar 

  • Foulis DJ, Pulmannová S (2012) Logical connectives on lattice effect algebras. Stud Logica 100(6):1291–1315

    Article  MathSciNet  Google Scholar 

  • Galatos N, Jipsen P, Kowalski T, Ono H (2007) Residuated lattices: an algebraic glimpse at substructural logics. Elsevier Science, Amsterdam

    MATH  Google Scholar 

  • Giuntini R (1996) Quantum MV algebras. Stud Logica 56:393–417

    Article  MathSciNet  Google Scholar 

  • Giuntini R, Ledda A, Paoli F (2016) A new view of effects in a Hilbert space. Stud Logica 104:1145–1177

    Article  MathSciNet  Google Scholar 

  • Giuntini R, Ledda A, Paoli F (2017) On Some Properties of \(PBZ^{\ast }\)-lattices. Int J Theor Phys 56:3895–3911

    Article  MathSciNet  Google Scholar 

  • Greechie R, Gudder S (1971) Is a quantum logic a logic? Helv Phys Acta 44:238–240

    MathSciNet  Google Scholar 

  • Greechie R, Gudder S, Reidel D (1973) Quantum logics. In: Hooker C (ed) Contemporary research in the foundations and philosophy of quantum theory. Kluwer Academic Publishers, Dordrect, pp 143–173

    Chapter  Google Scholar 

  • Hardegree GM (1981) Material implication in orthomodular (and boolean) lattices. Notre Dame J Form Logic 22(2):163–182

    Article  MathSciNet  Google Scholar 

  • Hooker CA (ed) (1975) The logico-algebraic approach to quantum mechanics, vol 1–2. D. Reidel Pub. Co, Dordrecht

    MATH  Google Scholar 

  • Jauch J, Piron C (1970) What is quantum logic. In: Freund P, Goebel C, Nabir Y (eds) Quanta. University of Chicago Press, Chicago, pp 842–848

    Google Scholar 

  • Kalman JA (1958) Lattices with antitone involution. Trans Am Math Soc 87:485–491

    Article  Google Scholar 

  • Kalmbach G (1983) Orthomodular lattices, vol III. Academic Press, London

    MATH  Google Scholar 

  • Olson MP (1971) The self-adjoint operators of a von Neumann algebra form a conditionally complete lattice. Proc Am Math Soc 28:537?544

    Article  Google Scholar 

  • Stroock DW (1998) A concise introduction to the theory of integration, 3rd edn. Birkhäuser, Basel

    Google Scholar 

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Acknowledgements

I. Chajda gratefully acknowledges IGA, Project PřF 2019 015. D. Fazio expresses his gratitude for the support of the Horizon 2020 program of the European Commission: SYSMICS Project, Number: 689176, MSCA-RISE-2015, and gratefully acknowledges Fondazione di Sardegna for its support within the project “Science and its Logics: The Representation’s Dilemma”, Cagliari, Number: F72F16003220002. Finally, the authors thank Francesco Paoli and Claudia Mureşan for their insightful suggestions about the issues addressed in this paper, and the anonymous referee for her helpful observations which have contributed to the improvement of this work.

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Authors and Affiliations

  1. Palacký University, Olomouc, Czech Republic

    I. Chajda

  2. University of Cagliari, Cagliari, Italy

    D. Fazio

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  1. I. Chajda
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  2. D. Fazio
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Correspondence to D. Fazio.

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Chajda, I., Fazio, D. On residuation in paraorthomodular lattices. Soft Comput 24, 10295–10304 (2020). https://doi.org/10.1007/s00500-020-04699-w

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