Weakly Distributive Domains

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 [D →D] fails ${\sf I}$. 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.