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Orthomodular Lattices in Occurrence Nets

  • Luca Bernardinello
  • Lucia Pomello
  • Stefania Rombolà
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5606)

Abstract

In this paper, we study partially ordered structures associated to occurrence nets. An occurrence net is endowed with a symmetric, but in general non transitive, concurrency relation. By applying known techniques in lattice theory, from any such relation one can derive a closure operator, and then an orthocomplemented lattice. We prove that, for a general class of occurrence nets, those lattices, formed by closed subsets of net elements, are orthomodular. A similar result was shown starting from a simultaneity relation defined, in the context of special relativity theory, on Minkowski spacetime. We characterize the closed sets, and study several properties of lattices derived from occurrence nets; in particular we focus on properties related to K-density. We briefly discuss some variants of the construction, showing that, if we discard conditions, and only keep the partial order on events, the corresponding lattice is not, in general, orthomodular.

Keywords

Closure Operator Maximal Clique Minkowski Spacetime Orthomodular Lattice Algebraic Lattice 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Petri, C.A.: Non-sequential processes. Technical Report ISF-77–5, GMD Bonn (1977) Translation of a lecture given at the IMMD Jubilee Colloquium on ‘Parallelism in Computer Science’, Universität Erlangen–Nürnberg (June 1976)Google Scholar
  2. 2.
    Best, E., Fernandez, C.: Nonsequential Processes–A Petri Net View. EATCS Monographs on Theoretical Computer Science, vol. 13. Springer, Heidelberg (1988)CrossRefzbMATHGoogle Scholar
  3. 3.
    Wolfram, S.: A New Kind of Science. Wolfram Media (2002)Google Scholar
  4. 4.
    Petri, C.A.: Rechnender netzraum. Spektrum der Wissenschaft, Spezial 3/07: Ist das Universum ein Computer? 16–19 (2007)Google Scholar
  5. 5.
    Petri, C.A.: On the physical basis of information flow – abstract. In: van Hee, K.M., Valk, R. (eds.) PETRI NETS 2008. LNCS, vol. 5062, p. 12. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  6. 6.
    Bombelli, L., Lee, J., Meyer, D., Sorkin, R.: Spacetime as a causal set. Phys. Rev. Lett. 60, 521–524 (1985)MathSciNetGoogle Scholar
  7. 7.
    Abramsky, S.: Petri nets, discrete physics, and distributed quantum computation. In: Degano, P., De Nicola, R., Meseguer, J. (eds.) Concurrency, Graphs and Models. LNCS, vol. 5065, pp. 527–543. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  8. 8.
    Cegła, W., Jadczyk, Z.: Causal logic of Minkowski space. Commun. Math. Phys. 57, 213–217 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Casini, H.: The logic of causally closed spacetime subsets. Class. Quantum Grav. 19, 6389–6404 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fernandez, C., Thiagarajan, P.S.: A lattice theoretic view of k-density. Arbeitspapiere der GMD, n. 76 (1983)Google Scholar
  11. 11.
    Nielsen, M., Plotkin, G.D., Winskel, G.: Petri nets, event structures and domains, part I. Theoretical Computer Science 13, 85–108 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Petri, C.A.: Nets, time and space. Theoretical Computer Science 153, 3–48 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Pták, P., Pulmannová, P.: Orthomodular Structures as Quantum Logics. Kluwer Academic Publishers, Dordrecht (1991)zbMATHGoogle Scholar
  14. 14.
    Beltrametti, E.G., Cassinelli, G.: The logic of quantum mechanics. Encyclopedia of Mathematics and its Applications, vol. 15. Addison-Wesley, Reading (1981)zbMATHGoogle Scholar
  15. 15.
    Birkhoff, G.: Lattice Theory, 3rd edn. American Mathematical Society (1979)Google Scholar
  16. 16.
    Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order. Cambridge University Press, Cambridge (1990)zbMATHGoogle Scholar
  17. 17.
    Petri, C.A.: Concepts of net theory. In: Mathematical Foundations of Computer Science: Proc. of Symposium and Summer School, High Tatras, September 3–8, 1973, pp. 137–146. Math. Inst. of the Slovak Acad. of Sciences (1973)Google Scholar
  18. 18.
    Kummer, O., Stehr, M.O.: Petri’s axioms of concurrency: A selection of recent results. In: Azéma, P., Balbo, G. (eds.) ICATPN 1997. LNCS, vol. 1248, pp. 195–214. Springer, Heidelberg (1997)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Luca Bernardinello
    • 1
  • Lucia Pomello
    • 1
  • Stefania Rombolà
    • 1
  1. 1.Dipartimento di informatica, sistemistica e comunicazioneUniversità degli studi di Milano–BicoccaMilanoItaly

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