A partial solution to an open problem of Amadio and Curien

Abstract

Amadio raised the question of whether the category of stable bifinite domains of Amadio–Droste is the largest Cartesian closed full subcategory of the category of ω-algebraic meet-cpos with stable functions. Zhang and Jiang showed that, for any ω-algebraic meet-cpo D, if the stable function space [D→sD] (with stable order) satisfies property M, then D is finitary (i.e. each compact element dominates only finitely many elements). In this paper, we show that, for any ω-algebraic meet-cpo D and the diamond lattice M (the classical non-distributive lattice), if [D→sM] and [[D→sM]→s[D→sM]] are ω-algebraic, then(1)D is finitary;(2)if [[D→sM]→s[D→sM]] is finitary, then D is stable bifinite. So the category of stable bifinite domains is a maximal Cartesian closed full subcategory of the category of ω-algebraic meet-cpos with stable functions. This gives a partial solution to the problem.