Effective cartesian closed categories of domains

Perhaps the most important and striking fact of domain theory is that important categories of domains are cartesian closed. This means that the category has a terminal object, finite products, and exponents. The only problematic part for domains is the exponent, which in this setting means the space of continuous functions. Cartesian closed categories of domains are well understood and the understanding is in some sense essentially complete by the work of Jung [5], Smyth [11], and others. In this paper we consider categories of effective domains. Again the function space is the crucial construction in order to obtain a cartesian closed category and we therefore concentrate on that. The case for effective algebraic domains is satisfactory. We introduce a natural notion of effective bifinite domains and show that the category of such is cartesian closed. This generalises the well-known construction for effective consistently complete algebraic cpos. The situation for continuous cpos is more problematic. One way to study effectivity on continuous cpos is to note that each continuous cpo is a projection of an algebraic cpo. Thus the more satisfactory theory of effectivity on algebraic cpos can be induced onto the continuous cpos via the projections. This was first done in Smyth [10] and consequently we use the term Smyth effective. Using the result for effective bifinite domains we prove that we can build Smyth effective type structures over Smyth effective continuous domains as long as these are projections of effective bifinite domains. The ability to build type structures of effective domains is important for several reasons. Such type structures carry a notion of effectiveness or computability which is given externally by recursive function theory via numberings. These computations can in principle be performed on a digital computer. External computations over type structures have applications, for example, in recursive analysis.