Cartesian Closed Categories of Algebraic CPOs

Abstract The results presented in this paper were inspired by the work of Smyth, who showed that there is a largest cartesian closed full subcategory inside the category of countably based algebraic cpos with least element, namely the category of profinite domains. Removing the countability condition, we show that there are exactly two maximal cartesian closed full subcategories inside the category of algebraic cpos with least element. One is the natural extension of the class of profinite domains to the uncountable case, the other is a new class of domains, which are characterized by the property that every principal ideal is a complete lattice. The name L- domain is introduced for cpos with this property. Passing to the general situation where no least element is required anymore, we find a pair of categories in place of each the profinite domains and the algebraic L-domains. Thus there are four maximal cartesian closed categories of algebraic cpos in this case. This complete overview over the possible classes of domains allows to prove general theorems about them. This is illustrated by the result that a cpo has an algebraic function space if and only if its space of strict continuous functions is algebraic.