Logically presented domains

We connect the theory of Scott-Ershov domains to first order model theory. The completeness property of domains is related to the model theoretic notion of saturation. In constraint programming this analogy is already used on the level of finite approximations. A simple relation to structures used in nonstandard analysis (ultra powers, Frechet powers) is obtained. This leads to natural logical presentations of domain constructions such as function space, products and the Smyth power domain. Sufficient conditions on models for constructing function spaces are given