Weakly Distributive Domains

  • Ying Jiang
  • Guo-Qiang Zhang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4583)


In our previous work [17] we have shown that for any ω-algebraic meet-cpo D, if all higher-order stable function spaces built from D are ω-algebraic, then D is finitary. This accomplishes the first of a possible, two-step process in solving the problem raised in [1,2]: whether the category of stable bifinite domains of Amadio-Droste-Göbel [1,6] is the largest cartesian closed full sub-category within the category of ω-algebraic meet-cpos with stable functions. This paper presents results on the second step, which is to show that for any ω-algebraic meet-cpo D satisfying axioms \({\sf M}\) and \({\sf I}\) to be contained in a cartesian closed full sub-category using ω-algebraic meet-cpos with stable functions, it must not violate \({\sf MI}^{\infty}\;\). We introduce a new class of domains called weakly distributive domains and show that for these domains to be in a cartesian closed category using ω-algebraic meet-cpos, property \({\sf MI}^{\infty}\;\) must not be violated. We further demonstrate that principally distributive domains (those for which each principle ideal is distributive) form a proper subclass of weakly distributive domains, and Birkhoff’s M 3 and N 5 [5] are weakly distributive (but non-distributive). We introduce also the notion of meet-generators in constructing stable functions and show that if an ω-algebraic meet-cpo D contains an infinite number of meet-generators, then [DD] fails \({\sf I}\). However, the original problem of Amadio and Curien remains open.


Stable Function Stable Domain Principal Ideal Stable Order Compact Element 
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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  1. Amadio, R.-M.: Bifinite domains: stable case. LNCS, vol. 530, pp. 16–33. Springer, Heidelberg (1991)Google Scholar
  2. Amadio, R.-M., Curien, P.-L.: Domains and Lambda-Calculi. Cambridge Tracts in Theeoretical Computer Science, vol. 46. Cambridge University Press, Cambridge (1998)zbMATHGoogle Scholar
  3. Berry, G.: Modèles complètement adéquats et stables des lambda-calculs typés. Thèse de Doctorat d’Etat, Université Paris VII (1979)Google Scholar
  4. Coquand, T., Gunter, C.A., Winskel, G.: DI-domains as a model of polymorphism. LNCS, vol. 298, pp. 344–363. Springer, Heidelberg (1987)Google Scholar
  5. Davey, B.-A., Priestley, H.-A.: Introduction to Lattices and Order. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar
  6. Droste, M.: On stable domains. Theoretical Computer Science 111, 89–101 (1993)zbMATHCrossRefGoogle Scholar
  7. Droste, M.: Cartesian closed categories of stable domains for polymorphism. Preprint, Universität GHS EssenGoogle Scholar
  8. Girard, J.-Y.: Linear logic. Theoretical Computer Science 50, 1–102 (1987)zbMATHCrossRefGoogle Scholar
  9. Milner, R.: Fully abstract models of typed λ-calculi. Theoretical Computer Science 4, 1–22 (1977)zbMATHCrossRefGoogle Scholar
  10. Plotkin, G.: A powerdomain construction. SIAM J. Computing 5, 452–487 (1976)zbMATHCrossRefGoogle Scholar
  11. Reddy, U.: Global state considered unnecessary: An introduction to object-based semantics. J. Lisp and Symbolic Computation 9, 7–76 (1996)CrossRefGoogle Scholar
  12. Smyth, M.B.: The largest cartesian closed category of domains. Theoretical Computer Science 27, 109–120 (1983)zbMATHCrossRefGoogle Scholar
  13. Winskel, G.: Linear Time, Branching Time and Partial Order in Logics and Models for Concurrency. LNCS, vol. 354, pp. 364–399. Springer, Heidelberg (1988)Google Scholar
  14. Zhang, G.-Q.: dI-domains as prime information systems. Information and Computation 100, 151–177 (1992)zbMATHCrossRefGoogle Scholar
  15. Zhang, G.-Q.: The largest cartesian closed category of stable domains. Theoretical Computer Science 166, 203–219 (1996)zbMATHCrossRefGoogle Scholar
  16. Zhang, G.-Q.: Logic of Domains. Birkhauser (1991)Google Scholar
  17. Zhang, G.-Q., Jiang, Y.: On a problem of Amadio and Curien: The finite antichain condition. Information and Computation 202, 87–103 (2005)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Ying Jiang
    • 1
  • Guo-Qiang Zhang
    • 2
  1. 1.State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, BeijingChina
  2. 2.Department of EECS, Case Western Reserve University, Cleveland, Ohio 44022USA

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