The Largest First-Order-Axiomatizable Cartesian Closed Category of Domains

The inspiration for this paper is a result proved by Michael Smyth which states that Gordon Plotkin's category SFP is the largest cartesian closed category of domains. Although this category is easily enough motivated from concepts in domain theory and category theory, it is clearly harder to describe and less \elementary" than the most popular categories of domains for denotational semantics. In particular, the category most often used by people who need domain theory is that of bounded complete algebraic cpo's. The use of this latter category has been championed by Dana Scott for years ([4], [5], [6]) and its use has become widespread. It is simple to describe, easy to work with, and su ces for most applications. The purpose of this paper is to state and prove an analog to Smyth's theorem which says that the bounded complete domains form the largest \easy to de ne" cartesian closed category of domains. Formally, a class K of domains will be \easy to de ne" if the posets which form the bases of members of K are the countable models of a rst order theory. This concentration on the bases is reasonable because many domains are best described by explaining what their compact elements are. The domain can then be constructed as the set of ideals of these compact elements ordered by set inclusion. (Indeed, this is a central idea urged in each of Scott's aforementioned papers.) Also, when working with domains, the compact elements are very handy for proving the kinds of facts that one needs to know. So it is important for the compact elements in the domains being used to lie in a familiar, easily-understood class. The second section of the paper discusses some of the de nitions and facts from domain theory and model theory which will be needed. In the third section we de ne precisely what is meant by a rst-order-axiomatizable class of domains and outline the proof of the main result. The proof uses Smyth's Theorem and the Compactness Theorem for rst order logic. An alternate proof using ultraproducts is also o ered. The nal section contains some discussion and a few remarks about possible extensions of the main result.