Quantum Logic in Dagger Kernel Categories

Order

Volumn 27, Issue 2, 2010, Pages 177-212

  Heunen, Chris ^a   Jacobs, Bart ^b  

^a UNIVERSITY OF OXFORD   (United Kingdom)

^b RADBOUD UNIVERSITY NIJMEGEN   (Netherlands)

Author keywords

Categorical logic; Dagger kernel category; Orthomodular lattice; Quantum logic

Indexed keywords

EID: 77953083086     PISSN: 01678094     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11083-010-9145-5     Document Type: Article

References (41)

1. Abramsky, S., Coecke, B.: A categorical semantics of quantum protocols. In: Logic in Computer Science, IEEE, pp. 415-425 (2004).
2. Awodey, S.: Category Theory. Oxford Logic Guides, Oxford Univ. Press, Oxford (2006).
3. Barr, M.: Algebraically compact functors. J. Pure Appl. Algebra 82, 211-231 (1992).
4. Barr, M., Wells, C.: Toposes, Triples and Theories. Springer, Berlin. Revised and corrected version available from URL: www.cwru.edu/artsci/math/wells/pub/ttt.html (1985).
5. Borceux, F.: Handbook of Categorical Algebra. Encyclopedia of Mathematics 50, 51 and 52. Cambridge Univ. Press, Cambridge (1994).
6. Cassinelli, G., De Vito, E. P., Lahti J., Levrero, A.: The Theory of Symmetry Actions in Quantum Mechanics. Number 654 in Lecture Notes in Physics. Springer, Berlin (2004).
7. Coecke, B., Pavlović, D.: Quantum measurements without sums. In: Chen, G., Kauffman, L., Lamonaco, S. (eds.) Mathematics of Quantum Computing and Technology. Taylor and Francis, Philadelphia. See also: arXiv:/quant-ph/0608035 (2006).
8. Coecke, B., Smets, S.: The Sasaki hook is not a [static] implicative connective but induces a backward [in time] dynamic one that assigns causes. Int. J. Theor. Phys. 43(7/8), 1705-1736 (2004).
9. Crown, G.: On some orthomodular posets of vector bundles. J. Nat. Sci. Math. 15(1-2), 11-25 (1975).
10. Davey, B., Priestley, H.: Introduction to Lattices and Order. Math. Textbooks. Cambridge Univ. Press, Cambridge (1990).
11. Dvurečenskij, A., Pulmannová, S.: New Trends in Quantum Structures. Kluwer Acad., Dordrecht (2000).
12. Finch, P. D.: Quantum logic as an implication algebra. Bull. Am. Math. Soc. 2, 101-106 (1970).
13. Foulis, D. J., Greechie, R., Bennett, M.: Sums and products of interval algebras. Int. J. Theor. Phys. 33(11), 2119-2136 (1994).
14. Freyd, P.: Abelian Categories: An Introduction to the Theory of Functors. Harper and Row, New York. Available via www.tac.mta.ca/tac/reprints/articles/3/tr3.pdf (1964).
15. Haghverdi, E., Scott, P.: A categorical model for the geometry of interaction. Theor. Comp. Sci. 350, 252-274 (2006).
16. Harding, J.: Orthomodularity of decompositions in a categorical setting. Int. J. Theor. Phys. 45(6), 1117-1128 (2006).
17. Heunen, C.: Compactly accessible categories and quantum key distribution. Logical Methods Comp. Sci. 4(4) (2008). doi: 10. 2168/LMCS-4(4: 9)2008.
18. Heunen, C.: Quantifiers for quantum logic. arXiv: 0811.1457 (2008).
19. Heunen, C.: An embedding theorem for Hilbert categories. Theory Appl. Categ. 22, 321-344 (2009).
20. Heunen, C.: Categorical quantum models and logics. Ph. D. thesis, Radboud Univ. Nijmegen (2010).
21. Jacobs, B.: Categorical Logic and Type Theory. North Holland, Amsterdam (1999).
22. Jacobs, B.: Orthomodular lattices, Foulis semigroups and dagger kernel categories. Logical Methods Comp. Sci. (2009). Available from http://arxiv.org/abs/0905.4090.
23. Janowitz, M. F.: Quantifiers and orthomodular lattices. Pac. J. Math. 13, 1241-1249 (1963).
24. Johnstone, P.: Stone Spaces. Number 3 in Cambridge Studies in Advanced Mathematics. Cambridge Univ. Press, Cambridge (1982).
25. Johnstone, P. T.: Sketches of an Elephant: A Topos Theory Compendium. Oxford University Press, Oxford (2002).
26. Kalmbach, G.: Orthomodular Lattices. Academic, London (1983).
27. Kalmbach, G.: Measures and Hilbert Lattices. World Scientific, Singapore (1986).
28. Kock, A., Reyes, G.: Doctrines in categorical logic. In: Barwise, J. (ed.) Handbook of Mathematical Logic, pp. 283-313. North-Holland, Amsterdam (1977).
29. Lane, S. M.: Categories for the Working Mathematician. Springer, Berlin (1971).
30. Lehmann, D.: A presentation of quantum logic based on an and then connective. J. Log. Comput. 18(1), 59-76 (2008).
31. Mac Lane, S.: An algebra of additive relations. Proc. Natl. Acad. sci. 47, 1043-1051 (1961).
32. Makkai, M., Reyes, G.: First Order Categorical Logic. Number 611 in Lect. Notes Math., Springer, Berlin (1977).
33. Manes, E.: Monads, matrices and generalized dynamic algebra. In: Ehrig, H., Herrlich, H., Kreowski, H.-J., Preuß, G. (eds.) Categorical Methods in Computer Science With Aspects from Topology, number 393 in Lect. Notes Comp. Sci, pp. 66-81 (1989).
34. Piron, C.: Foundations of Quantum Physics. Number 19 in Mathematical Physics Monographs. Benjamin, New York (1976).
35. Puppe, D.: Korrespondenzen in abelschen Kategorien. Math. Ann. 148, 1-30 (1962).
36. Román, L.: A characterization of quantic quantifiers in orthomodular lattices. Theory Appl. Categ. 16, 206-217 (2006).
37. Román, L., Rumbos, B.: A characterization of nuclei in orthomodular and quantic lattices. J. Pure Appl. Algebra 73, 155-163 (1991).
38. Román, L., Zuazua, R.: On quantic conuclei in orthomodular lattices. Theory Appl. Categ. 2(6), 62-68 (1996).
39. Rosenthal, K.: Quantales and Their Applications. Number 234 in Pitman Research Notes in Math. Longman Scientific & Technical, Harlow (1990).
40. Selinger, P.: Dagger compact closed categories and completely positive maps (extended abstract). In: Selinger, P. (ed.) Proceedings of the 3rd International Workshop on Quantum Programming Languages (QPL 2005), number 170 in Elect. Notes in Theor. Comp. Sci., pp. 139-163. http://dx.doi.org/10.1016/j.entcs.2006.12.018 (2007).
41. Taylor, P.: Practical Foundations of Mathematics. Number 59 in Cambridge Studies in Advanced Mathematics. Cambridge Univ. Press, Cambridge (1999).