On the largest Cartesian closed category of stable domains

Abstract

Let D be a Scott-domain and [D→cD] (resp., [D→sD]) its conditionally multiplicative (CM for short) (resp., stable) function space. Zhang (1996) [19] mentioned that if [D→cD] is bounded complete, D should be distributive. In the first part of this paper, we prove that if [D→cD] or [D→sD] is bounded complete, then D is distributive, which confirms that his conjecture is true.Amadio (1991) [3] and Curien (1998) [4] raised the question of whether the category of stable bifinite domains (SB for short) in sense of Amadio–Droste is the largest Cartesian closed full subcategory of the category of ω-algebraic meet-cpos with CM functions (ω-SAM for short). In the second part of this paper, we prove that for any ω-algebraic meet-cpo D and certain non-distributive finite poset M˜, if [D→cM˜], [[D→cM˜]→c[D→cM˜]] and [[[D→cM˜]→cM˜]→c[[D→cM˜]→cM˜]] are ω-algebraic, then we have that (1) D is finitary; (2) if D is not stable bifinite, then [[D→cM˜]→c[D→cM˜]] is not finitary. So, the category SB is a maximal Cartesian closed full subcategory of ω-SAM, which gives a partial solution to the problem posed by Amadio and Curien.