• Jan Paseka
  • Jiří Rosický
Part of the Fundamental Theories of Physics book series (FTPH, volume 111)


In this paper we survey aspects of the theory of quantales, starting from its connec-tion to locales and C*-algebras and finishing with recent developments involving the notions simplicity and spatiality.


Residuated Lattice Frame Homomorphism Spatial Quantales Dual Atom Lambek Calculus 
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  1. [1]
    Akemann, C.A. (1970) Left ideal structure of Calgebras, Journal of Functional Analysis 6, 305–317.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    Akemann, C.A. (1970) A Gelfand representation theory of Calgebras, Pacific Journal of Mathematics 6, 305–317.MathSciNetzbMATHGoogle Scholar
  3. [3]
    Borceux, F. and Van Den Bossche, G. (1989) An essay on noncommutative topology, Topology and its Applications 31, 203–223.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    Borceux, F., Rosickÿ, R. and Van Den Bossche, G. (1989) Quantales and C-algebras, Journal of the London Mathematical Society 40, 398–404.zbMATHCrossRefGoogle Scholar
  5. [5]
    Dilworth, R.P. (1939) Non-commutative residuated lattices, Transactions of the American Mathematical Society 46, 426–444.MathSciNetGoogle Scholar
  6. [6]
    Dixmier, J. (1977) Calgebras, North Holland, Amsterdam.Google Scholar
  7. [7]
    Giles, R. and Kummer, H. (1971) A non-commutative generalization of topology, Indiana University Mathematical Jounal 21, 91–102.MathSciNetCrossRefGoogle Scholar
  8. [8]
    Girard, J.-Y (1997) Linear logic, Theoretical Computer Science, 50, 1–102.CrossRefGoogle Scholar
  9. [9]
    Johnstone, P.T. (1982) Stone Spaces, Cambridge UP, Cambridge.zbMATHGoogle Scholar
  10. [10]
    Joyal, A. and Tierney, M. (1984) An Extension of the Galois Theory of Grothendieck, Memoirs of the American Mathematical Society 309.Google Scholar
  11. [11]
    Kadison, R. and Ringrose, J. (1997) Fundamentals of the Theory of Operator Algebras, Academic Press, New York.Google Scholar
  12. [12]
    Krull, W. (1924) Axiomatische begründung der allgemeinen idealtheorie, Sitzungsberichte der Physikalisch-Medicinischen Sociatät zu Erlangen 56, 4763.Google Scholar
  13. [13]
    Kruml, D. (2000) Spatial quantales, Applied Categorical Structures, To appear.Google Scholar
  14. [14]
    Mulvey, C.J. (1986), Supplemento ai Rendiconti del Circolo Matematico di Palermo II 12, 99–104.MathSciNetGoogle Scholar
  15. [15]
    Mulvey, C.J. and Wick Pelletier, J. (1992) A quantisation of the calculus of relations, Canadian Mathematical Society Conference Proceeding 13, 345–360.Google Scholar
  16. [16]
    Mulvey, C.J. and Wick Pelletier, J. (n.d.) On the quantisation of points, To appear.Google Scholar
  17. [17]
    Paseka, J. (1997) Simple quantales, Proceedings of the Eighth Prague Topological Symposium 1996, (Topology Atlas 1997), 314–328.MathSciNetGoogle Scholar
  18. [18]
    Paseka, J. and Kruml, D. (2000) Embeddings of quantales into simple quantales, Journal of Pure and Applied Algebra, To appear.Google Scholar
  19. [19]
    Wick Pelletier, J. and Rosickÿ, J. (1997) Simple involutive quantales, Journal of Algebra 195, 367–386.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    Rosenthal, K.I. (1990) Quantales and Their Applications, Pitman Research Notes in Mathematics Series 234, Longman.Google Scholar
  21. [21]
    Rosickÿ, J. (1989) Multiplicative lattices and C*-algebras, Cahiers de Topologie et Géométrie Différentielle Catégoriques 30, 95–110.zbMATHGoogle Scholar
  22. [22]
    RosickSr, J. (1995) Characterizing spatial quantales, Algebra Universalis 34, 175–178.MathSciNetCrossRefGoogle Scholar
  23. [23]
    Vickers, S. (1989) Topology via Logic, Cambridge UP, Cambridge.zbMATHGoogle Scholar
  24. [24]
    Ward, M. (1937) Residuations in structures over which a multiplication is defined, Duke Mathematical Journal 3, 627–636.MathSciNetCrossRefGoogle Scholar
  25. [25]
    Ward, M. (1938) Structure residuation, Annals of Mathematics 39, 558–568.MathSciNetCrossRefGoogle Scholar
  26. [26]
    Ward, M. and Dilworth, R.P. (1939) Residuated lattices, Transactions of the American Mathematical Society 45, 335–354.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • Jan Paseka
    • 1
  • Jiří Rosický
    • 1
  1. 1.Department of MathematicsMasaryk UniversityBrnoCzech Republic

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