On an open problem of Amadio and Curien: The finite antichain condition

More than a dozen years ago, Amadio [1] (see Amadio and Curien [2] as well) raised the question of whether the category of stable bifinite domains of Amadio-Droste [1,6,7] is the largest cartesian closed full sub-category of the category of ω-algebraic meet-cpos with stable functions. A solution to this problem has two major steps: (1) Show 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 (i.e., it satisfies the so-called axiom I); (2) Show that for any ω-algebraic meet-cpo D, if D violates MI , then [D → D] violates either M or I. We solve the first part of the problem in this paper, i.e., for any ω-algebraic meet-cpo D, if the stable function space [D → D] satisfies M, then D is finitary. Our notion of (mub, meet)-closed set, which is introduced for step 1, will also be used for treating some example cases in step 2.