Finite Axiomatizations for Universal Domains

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Abstract
In the theory of denotational semantics of programming languages, several authors established the existence of particular kinds of universal domains. Here we consider the categories of all ω-bifinite domain, all ω-bifinite L-domains, all ω-Scott-domains, and all ω-algebraic lattices, respectively, in each case with embedding-projection pairs as morphisms. It has been shown that each of these categories contains a universal homogeneous, or saturated. object. which is unipue up to isomorphism. Here we introduce for each of these four categories l a finite set of axioms Sl, formulated in a first-order language of predicate calculus for posets, and show that an arbitrary domain (D, ≤)ε l is the universal homogeneous object in l if and only if its subset of compact elements satisfies all axioms in Sl.

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