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Daggers, Kernels, Baer *-semigroups, and Orthomodularity

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Abstract

We discuss issues related to constructing an orthomodular structure from an object in a category. In particular, we consider axiomatics related to Baer *-semigroups, partial semigroups, and various constructions involving dagger categories, kernels, and biproducts.

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References

  1. Abramsky, S., & Coecke, B. (2004). A categorical semantics of quantum protocols. In Proceedings 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).

  2. Barnum, H., & Wilce, A. (2009). Ordered linear spaces and categories as frameworks for information-processing characterizations of quantum and classical theory (preprint).

  3. Beran, L. (1985). Orthomodular lattices: algebraic approach. Dordrecht: Reidel Publishing Company.

    Book  Google Scholar 

  4. Berberian, S.K. (1972). Baer *-rings, Die Grundlehren der mathematischen Wissenschaften band 195. New York-Berlin: Springer-Verlag.

    Google Scholar 

  5. Coecke, B., & Duncan, R. (2011). Interacting quantum observables: Categorical algebra and diagrammatics. New Journal of Physics, 13, 85.

    Google Scholar 

  6. Crown, G.D. (1975). Some orthomodular posets of vector bundles. Nature Science and Mathematics, 15(1–2), 11–25.

    Google Scholar 

  7. Dalla Chiara, M.L., Giuntini, R., Greechie, R. (2004). Reasoning in quantum theory: sharp and unsharp quantum logics. Dordrecht: Kluwer.

    Book  Google Scholar 

  8. Dvurećenskij, A. (1993). Gleason’s theorem and its applications. Mathematics and its applications (East European Series) (Vol. 60). Dordrecht: Kluwer.

    Book  Google Scholar 

  9. Foulis, D.J. (1960). Baer *-semigroups. Proceedings of the American Mathematical Society, 11, 648–654.

    Google Scholar 

  10. Foulis, D.J. (1961). Conditions for the modularity of an orthomodular lattice. Pacific Journal of Mathematics, 11, 889–895.

    Article  Google Scholar 

  11. Foulis, D.J. (1963). Relative inverses in Baer *-semigroups. Michigan Mathematical Journal, 10, 65–84.

    Article  Google Scholar 

  12. Foulis, D.J. (1965). Semigroups coordinatizing orthomodular geometries. Canadian Journal of Mathematics, 17, 40–51.

    Article  Google Scholar 

  13. Foulis, D.J. (1968). Multiplicative elements in Baer *-semigroups. Mathematische Annalen, 175, 297–302.

    Article  Google Scholar 

  14. Greechie, R.J. (2009). Oral communication.

  15. Gudder, S.P. (1972). Partial algebraic structures associated with orthomodular posets. Pacific Journal of Mathematics, 41, 717–730.

    Article  Google Scholar 

  16. Gudder, S.P., & Schelp, R.H. (1970). Coordinatization of orthocomplemented and orthomodular posets. Proceedings of the American Mathematical Society, 25, 229–237.

    Article  Google Scholar 

  17. Harding, J. (1996). Decompositions in quantum logic. Transactions of the American Mathematical Society, 348(5), 1839–1862.

    Article  Google Scholar 

  18. Harding, J. (2006). Orthomodularity of decompositions in a categorical setting. International Journal of Theoretical Physics, 45(6), 1117–1128.

    Article  Google Scholar 

  19. Harding, J. (2009). A link between quantum logic and categorical quantum mechanics. International Journal of Theoretical Physics, 48(3), 769–802.

    Article  Google Scholar 

  20. Herrlich, H., & Strecker, G. (1973). Category theory: an introduction. Allyn and Bacon.

  21. Heunen, C., & Jacobs, B. (2009). Quantum logic in dagger kernel categories, to appear. In B. Coecke, P. Panangaden & P. Selinger (Eds.), Proceedings of the 6th international workshop on quantum programming languages (QPL2009).. Elect Notes in Theor. Comp. Sci. Amsterdam: Elsevier. Available from: http://arxiv.org/abs/0902.2355.

  22. Jacobs, B. (2010). Orthomodular lattices, Foulis semigroups, and dagger kernel categories. Logical Methods in Computer Science, 6(2:1), 26.

    Google Scholar 

  23. Kalmbach, G. (1983). Orthomodular lattices. London mathematical society monographs (Vol. 18). London: Academic Press.

    Google Scholar 

  24. Karoubi, M. (2008). K-Theory: an introduction. Heidelberg: Springer-Verlag.

    Google Scholar 

  25. Kuhn, K. (1983). Extending homomorphisms from orthomodular lattices to Foulis semigroups. Contributions to General Algebra, 2 (Klagenfurt, 1982) (pp. 229–232). Vienna: Hölder-Pichler-Tempsky.

    Google Scholar 

  26. Maeda, F., & Maeda, S. (1970). Theory of symmetric lattices. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Band 173. New York-Berlin: Springer-Verlag.

    Google Scholar 

  27. Mac Lane, S. (1971). Categories for the working mathematician, Springer graduate texts in mathematics (Vol. 5). New York: Springer.

  28. Pták, P., & Pulmanová, S. (1991). Orthomodular structures as quantum logics. Fundamental theories of physics (Vol. 44). Dordrecht: Kluwer.

  29. Selinger, P. (2007). Dagger compact closed categories and completely positive maps (extended abstract). Electronic Notes in Theoretical Computer Science, 170, 139–163.

    Article  Google Scholar 

  30. Selinger, P. (2008). Idempotents in dagger categories (extended abstract). Electronic Notes in Theoretical Computer Science, 210, 107–122.

    Article  Google Scholar 

  31. Skornyakov, L.A. (1964). Complemented modular lattices and regular rings. Edinburgh-London: Oliver and Boyd.

    Google Scholar 

  32. von Neumann, J. (1960). Continuous geometry. Princeton: Princeton University Press.

    Google Scholar 

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Harding, J. Daggers, Kernels, Baer *-semigroups, and Orthomodularity. J Philos Logic 42, 535–549 (2013). https://doi.org/10.1007/s10992-013-9275-5

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