On the Largest Cartesian Closed Category of Stable Domains

Let SABC (resp., SABC˜) be the category of algebraic bounded complete domains with conditionally multiplicative mappings, that is, Scott-continuous mappings preserving meets of pairs of compatible elements (resp., stable mappings). Zhang showed that the category of dI-domains is the largest cartesian closed subcategory of ω-SABC and ω-SABC˜, with the exponential being the stable function space, where ω-SABC and ω-SABC˜ are full subcategories of SABC and SABC˜ respectively which contain countablly based algebraic bounded complete domains as objects. This paper shows that: The exponentials of any full subcategory of SABC or SABC˜ are exactly function spaces; SDABC˜ the category of distributive algebraic bounded complete domains, is the largest cartesian closed subcategory of SABC˜;