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Weakly Distributive Domains

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

Abstract

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.

Keywords

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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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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