Weakly distributive domains (II)

Abstract

In our previous work (Inform. and Comput., 2005, 202: 87–103), 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 (LNCS, 1991, 530: 16–33; Domains and lambda-calculi, Cambridge Univ. Press, 1998) whether the category of stable bifinite domains of Amadio-Droste-Göbel (LNCS, 1991, 530: 16–33; Theor. Comput. Sci., 1993, 111: 89–101) is the largest cartesian closed full subcategory within the category of ω-algebraic meet-cpos with stable functions. This paper presents the results of the 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. Further, we 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 (Introduction to Lattices and order, Cambridge Univ. Press, 2002) are weakly distributive (but non-distributive). Then, we establish characterization results for weakly distributive domains. We also introduce 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.