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Piecewise Boolean Algebras and Their Domains

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Automata, Languages, and Programming (ICALP 2014)

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Abstract

We characterise piecewise Boolean domains, that is, those domains that arise as Boolean subalgebras of a piecewise Boolean algebra. This leads to equivalent descriptions of the category of piecewise Boolean algebras: either as piecewise Boolean domains equipped with an orientation, or as full structure sheaves on piecewise Boolean domains.

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References

  1. van den Berg, B., Heunen, C.: Noncommutativity as a colimit. Appl. Cat. Struct. 20(4), 393–414 (2012)

    MATH  Google Scholar 

  2. Hailperin, T.: Boole’s algebra isn’t Boolean algebra. Math. Mag. 54(4) (1981)

    Google Scholar 

  3. Hughes, R.I.G.: Omnibus review. J. Symb. Logic 50(2), 558–566 (1985)

    Article  Google Scholar 

  4. Finch, P.E.: On the structure of quantum logic. J. Symb. Logic 34(2) (1969)

    Google Scholar 

  5. Gudder, S.P.: Partial algebraic structures associated with orthomodular posets. Pacific J. Math. 41(3) (1972)

    Google Scholar 

  6. Kalmbach, G.: Orthomodular Lattices. Acad. Pr (1983)

    Google Scholar 

  7. Abramsky, S., Jung, A.: Domain Theory. In: Handbook of Logic in Comp. Sci., vol. 3. Clarendon Press (1994)

    Google Scholar 

  8. Jung, A.: Cartesian closed categories of domains. PhD thesis, Tech. Hochsch. Darmstadt (1988)

    Google Scholar 

  9. Heunen, C., Landsman, N.P., Spitters, B.: A topos for algebraic quantum theory. Comm. Math. Phys. 291, 63–110 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  10. Döring, A., Barbosa, R.S.: Unsharp values, domains and topoi. In: Quantum Field Theory and Gravity, pp. 65–96. Birkhäuser (2011)

    Google Scholar 

  11. Harding, J., Navara, M.: Subalgebras of orthomodular lattices. Order 28, 549–563 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  12. Grätzer, G., Koh, K.M., Makkai, M.: On the lattice of subalgebras of a Boolean algebra. Proc. Amer. Math. Soc. 36, 87–92 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  13. Sachs, D.: The lattice of subalgebras of a Boolean algebra. Can. J. Math. 14, 451–460 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  14. Johnstone, P.T.: Stone spaces. Cambridge Studies in Advanced Mathematics, vol. 3. Cambridge Univ. Pr. (1982)

    Google Scholar 

  15. Laird, J.: Locally Boolean domains. Theor. Comp. Sci. 342(1), 132–148 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  16. Abramsky, S., Vickers, S.: Quantales, observational logic and process semantics. Math. Struct. Comp. Sci. 3, 161–227 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  17. Scott, D.S.: Domains for denotational semantics. In: Nielsen, M., Schmidt, E.M. (eds.) ICALP 1982. LNCS, vol. 140, pp. 577–613. Springer, Heidelberg (1982)

    Chapter  Google Scholar 

  18. Haimo, F.: Some limits of Boolean algebras. Proc. Amer. Math. Soc. 2(4), 566–576 (1951)

    Article  MATH  MathSciNet  Google Scholar 

  19. Sachs, D.: Partition and modulated lattices. Pacific J. Math. 11(1), 325–345 (1961)

    Article  MATH  MathSciNet  Google Scholar 

  20. Sasaki, U., Fujiwara, S.: The characterization of partition lattices. J. Sci. Hiroshima Univ (A) 15, 189–201 (1952)

    MATH  MathSciNet  Google Scholar 

  21. Ore, O.: Theory of equivalence relations. Duke Math. J. 9(3), 573–627 (1942)

    Article  MATH  MathSciNet  Google Scholar 

  22. Firby, P.A.: Lattices and compactifications I, II. Proc. London Math. Soc. 27, 22–60 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  23. Aigner, M.: Uniformität des Verbandes der Partitionen. Math. Ann. 207, 1–22 (1974)

    Article  MathSciNet  Google Scholar 

  24. Stonesifer, J.R., Bogart, K.P.: Characterizations of partition lattices. Alg. Univ. 19, 92–98 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  25. Yoon, Y.J.: Characterizations of partition lattices. Bull. Korean Math. Soc. 31(2), 237–242 (1994)

    MATH  MathSciNet  Google Scholar 

  26. Koppelberg, S.: Handbook of Boolean algebras, vol. 1. North-Holland (1989)

    Google Scholar 

  27. Cannon, S.: The spectral presheaf of an orthomodular lattice. Master’s thesis, Univ. Oxford (2013)

    Google Scholar 

  28. Harding, J., Döring, A.: Abelian subalgebras and the Jordan structure of a von Neumann algebra. Houston J. Math (2014)

    Google Scholar 

  29. Hamhalter, J.: Isomorphisms of ordered structures of abelian C*-subalgebras of C*-algebras. J. Math. Anal. Appl. 383, 391–399 (2011)

    Article  MATH  MathSciNet  Google Scholar 

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Heunen, C. (2014). Piecewise Boolean Algebras and Their Domains. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds) Automata, Languages, and Programming. ICALP 2014. Lecture Notes in Computer Science, vol 8573. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43951-7_18

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  • DOI: https://doi.org/10.1007/978-3-662-43951-7_18

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-43950-0

  • Online ISBN: 978-3-662-43951-7

  • eBook Packages: Computer ScienceComputer Science (R0)

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