Equivalence Of The Algebra L-domain Representation Theorem And Its Related Areas

Domain theory plays a fundamental role in denotational semantics of programming languages. Its importance may be judged from the fact that it has had many applications in fields as diverse as general topology, lattice theory, category theory and theoretical computer science as well as in many other areas of mathematics. Firstly, this paper introduces the new concepts of locally conditional upper semilattice (in short, L-cusl) and its ideal completion. With these concepts, the following results are obtained: (1) the set of compact elements in an algebraic L-domain is an L-cusl; (2) algebraic L-domains equipped with Scott topology are the soberification of their sets of compact elements equipped with pseudoScott topology; (3) the ideal completion of an L-cusl is an algebraic L-domain, obtaining a representation theorem for algebraic L-domains. It is also proved that the category of algebraic L-domains with scott continuous functions as morphisms is a reflective subcategory of the category of L-cusls and monotone maps. Secondly, based on the theory of consistent L-domains and their directed completions established by professor Xu, this paper proves that the set of compact elements in a consistently algebraic L-domain is an L-cusl and that the ideal completion of the set of compact elements in a consistently algebraic L-domain is isomorphic to its directed completion, which characterizes the directed completion of consistently algebraic L-domains. Finally, this paper introduces the concepts of subL-cusl, algebraic subL-domain, L-cusl embedding and projection pair for domains. Making use of the representation theorem for algebraic L-domains, this paper proves that the category of algebraic L-domains with projection pairs for domains as morphisms is eqivalent to the category of L-cusls and L-cusl embeddings.