A partial solution to an open problem of Amadio and Curien

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