Piecewise Boolean Algebras and Their Domains

  • Chris Heunen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8573)


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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  1. 1.
    van den Berg, B., Heunen, C.: Noncommutativity as a colimit. Appl. Cat. Struct. 20(4), 393–414 (2012)MATHGoogle Scholar
  2. 2.
    Hailperin, T.: Boole’s algebra isn’t Boolean algebra. Math. Mag. 54(4) (1981)Google Scholar
  3. 3.
    Hughes, R.I.G.: Omnibus review. J. Symb. Logic 50(2), 558–566 (1985)CrossRefGoogle Scholar
  4. 4.
    Finch, P.E.: On the structure of quantum logic. J. Symb. Logic 34(2) (1969)Google Scholar
  5. 5.
    Gudder, S.P.: Partial algebraic structures associated with orthomodular posets. Pacific J. Math. 41(3) (1972)Google Scholar
  6. 6.
    Kalmbach, G.: Orthomodular Lattices. Acad. Pr (1983)Google Scholar
  7. 7.
    Abramsky, S., Jung, A.: Domain Theory. In: Handbook of Logic in Comp. Sci., vol. 3. Clarendon Press (1994)Google Scholar
  8. 8.
    Jung, A.: Cartesian closed categories of domains. PhD thesis, Tech. Hochsch. Darmstadt (1988)Google Scholar
  9. 9.
    Heunen, C., Landsman, N.P., Spitters, B.: A topos for algebraic quantum theory. Comm. Math. Phys. 291, 63–110 (2009)CrossRefMATHMathSciNetGoogle Scholar
  10. 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. 11.
    Harding, J., Navara, M.: Subalgebras of orthomodular lattices. Order 28, 549–563 (2011)CrossRefMATHMathSciNetGoogle Scholar
  12. 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)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Sachs, D.: The lattice of subalgebras of a Boolean algebra. Can. J. Math. 14, 451–460 (1962)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Johnstone, P.T.: Stone spaces. Cambridge Studies in Advanced Mathematics, vol. 3. Cambridge Univ. Pr. (1982)Google Scholar
  15. 15.
    Laird, J.: Locally Boolean domains. Theor. Comp. Sci. 342(1), 132–148 (2005)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Abramsky, S., Vickers, S.: Quantales, observational logic and process semantics. Math. Struct. Comp. Sci. 3, 161–227 (1993)CrossRefMATHMathSciNetGoogle Scholar
  17. 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)CrossRefGoogle Scholar
  18. 18.
    Haimo, F.: Some limits of Boolean algebras. Proc. Amer. Math. Soc. 2(4), 566–576 (1951)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Sachs, D.: Partition and modulated lattices. Pacific J. Math. 11(1), 325–345 (1961)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Sasaki, U., Fujiwara, S.: The characterization of partition lattices. J. Sci. Hiroshima Univ (A) 15, 189–201 (1952)MATHMathSciNetGoogle Scholar
  21. 21.
    Ore, O.: Theory of equivalence relations. Duke Math. J. 9(3), 573–627 (1942)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Firby, P.A.: Lattices and compactifications I, II. Proc. London Math. Soc. 27, 22–60 (1973)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Aigner, M.: Uniformität des Verbandes der Partitionen. Math. Ann. 207, 1–22 (1974)CrossRefMathSciNetGoogle Scholar
  24. 24.
    Stonesifer, J.R., Bogart, K.P.: Characterizations of partition lattices. Alg. Univ. 19, 92–98 (1984)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Yoon, Y.J.: Characterizations of partition lattices. Bull. Korean Math. Soc. 31(2), 237–242 (1994)MATHMathSciNetGoogle Scholar
  26. 26.
    Koppelberg, S.: Handbook of Boolean algebras, vol. 1. North-Holland (1989)Google Scholar
  27. 27.
    Cannon, S.: The spectral presheaf of an orthomodular lattice. Master’s thesis, Univ. Oxford (2013)Google Scholar
  28. 28.
    Harding, J., Döring, A.: Abelian subalgebras and the Jordan structure of a von Neumann algebra. Houston J. Math (2014)Google Scholar
  29. 29.
    Hamhalter, J.: Isomorphisms of ordered structures of abelian C*-subalgebras of C*-algebras. J. Math. Anal. Appl. 383, 391–399 (2011)CrossRefMATHMathSciNetGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Chris Heunen
    • 1
  1. 1.Department of Computer ScienceUniversity of OxfordUnited Kingdom

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