Weakly Distributive Domains [chapter]

Ying Jiang, Guo-Qiang Zhang
2007 Lecture Notes in Computer Science  
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 subcategory within the category of ω-algebraic meet-cpos with stable functions. This paper presents results on
more » ... e second step, which is to show that for any ω-algebraic meet-cpo D satisfying axioms M and I to be contained in a cartesian closed full sub-category using ω-algebraic meet-cpos with stable functions, it must not violate MI ∞ . 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 MI ∞ 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 [D → D] fails I. However, the original problem of Amadio and Curien remains open. ⋆ This work is partially supported by NSFC 60673045, NSFC 60373050, NSFC major research program 60496321 and NSFC 60421001. If D is a weakly distributive, ω-algebraic meet-cpo with properties M and I, but not MI ∞ , then an infinite A as mentioned in Theorem 3 exists. Moreover, since D is a weakly distributive, every element of A is a generator. Therefore, we have the following corollary. Theorem 4. Let D be a weakly distributive, ω-algebraic meet-cpo with properties M and I, but not MI ∞ . Then [D → D] fails I.
doi:10.1007/978-3-540-73228-0_15 fatcat:xvl4t2mkbrff3dz7uqk5s5xne4