Coherence and consistency in domains
Carl A. Gunter, Achim Jung · Journal of Pure and Applied Algebra · 1990
Almost all of the categories normally used as a mathematical foundation for denotational semantics satisfy a condition known as consistent completeness. The goal of this paper is to explore the possibility of using a different condition - that of coherence - which has its origins in topology and logic. In particular, we concentrate on those posets whose principal ideals are algebraic lattices and whose topologies are coherent. These form a Cartesian closed category which has fixed points for domain equations. It is shown that a ‘universal domain’ exists. Since the construction of this domain seems to be of general significance, a categorical treatment is provided and applied to other classes of domains. Universal domains constructed in this fashion enjoy an additional property: they are saturated. We show that there is exactly one such domain in each of the classes under consideration.