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Quantum Logic, Dagger Kernel Categories and Inverse Baer*-Categories

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Abstract

This paper investigates dagger kernel categories which are considered first by Crown (J Nat Sci Math 15:11–25, 1975) and used by Heunen and Jacobs (Order 27:177–212, 2010) in their study of quantum logic from the perspective of categorical logic. The inverse Baer*-categories with splitting projections as special dagger kernel categories have a central place in our investigations. The inverse Baer*-categories with splitting and closed projections are Boolean and therefore the subobject lattices of such categories are representing classical logics. Examples are presented at every stage of our investigations.

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References

  1. Abramsky, S., Coecke, B.: A categorical semantics of quantum protocols. In: Proc. of the 19th Annual IEEE Symposium on Logic in Computer Science (LiCS 04), pp. 415–425. IEEE Computer Science Press. Extended version at arXiv:quant-ph/0402130 (2004)

  2. Crown, G.D.: Some orthomodular posets of vector bundles. J. Nat. Sci. Math. 15, 11–25 (1975)

    MathSciNet  MATH  Google Scholar 

  3. Foulis, D.J.: Baer*-semigroups. Proc. Am. Math. Soc. 11, 648–654 (1960)

    MathSciNet  MATH  Google Scholar 

  4. Foulis, D.J.: Relative inverses in Baer*-semigroups. Mich. Math. J. 10, 65–84 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  5. Harding, J.: Orthomodularity of decompositions in a categorical setting. Int. J. Theor. Phys. 45, 1117–1128 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Harding, J.: A link between quantum logic and categorical quantum mechanics. Int. J. Theor. Phys. 48, 769–802 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Harding, J.: Daggers, kernels, Baer*-semigroups, and orthomodularity. http://isis.nmsu.edu/~jharding/2010%20daggers%20kernels%20Baer%20orthomodularity.pdf. Accessed 24 March 2011

  8. Heunen, C.: Categorical Quantum Models and Logics. Pallas Publ., Amsterdam University Press (2009)

  9. Heunen, C., Jacobs, B.: Quantum logic in dagger kernel categories. Order 27, 177–212 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Jacobs, B.: Orthomodular lattices, Foulis semigroups and dagger kernel categories. In: Logical Methods in Comp. Sci. Available from http://arxiv.org/abs/0905.4090 (2009)

  11. Kastl, J.: Inverse Categories, Studien zur Algebra und ihre Anwendungen, Band 7, pp. 51–60. Akademie, Berlin (1979)

    Google Scholar 

  12. Mitchell, B.: Theory of Categories. Academic, New York (1965)

    MATH  Google Scholar 

  13. Schwab, E.: Uber das verallgemeinerte inverse eines Morphismus. Math. Nachr. 81, 243–257 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  14. Schwab, E.: Image and inverse image mappings in exact inverse categories. Boll. U.M.I. 18-B, 832–845 (1981)

    MathSciNet  Google Scholar 

  15. Schwab, E., Schwab, E.D.: The inverse Baer*-category of a chain. Rad. Mat. 11, 7-11 (2002)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. West University of Timisoara, Timisoara, Romania

    Emil Schwab

  2. University of Texas at El Paso, El Paso, TX, 79902, USA

    Emil Daniel Schwab

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  1. Emil Schwab
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  2. Emil Daniel Schwab
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Correspondence to Emil Daniel Schwab.

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Schwab, E., Schwab, E.D. Quantum Logic, Dagger Kernel Categories and Inverse Baer*-Categories. Order 29, 405–417 (2012). https://doi.org/10.1007/s11083-011-9211-7

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  • DOI: https://doi.org/10.1007/s11083-011-9211-7

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