# Effective Domains and Admissible Domain Representations

Abstract: This thesis consists of four papers in domain theory and a summary. The first two papers deal with the problem of defining effectivity for continuous cpos. The third and fourth paper present the new notion of an admissible domain representation, where a domain representation D of a space X is ?-admissible if, in principle, all other ?-based domain representations E of X can be reduced to X via a continuous function from E to D. In Paper I we define a cartesian closed category of effective bifinite domains. We also investigate the method of inducing effectivity onto continuous cpos via projection pairs, resulting in a cartesian closed category of projections of effective bifinite domains. In Paper II we introduce the notion of an almost algebraic basis for a continuous cpo, showing that there is a natural cartesian closed category of effective consistently complete continuous cpos with almost algebraic bases. We also generalise the notion of a complete set, used in Paper I to define the bifinite domains, and investigate what closure results that can be obtained. In Paper III we consider admissible domain representations of topological spaces. We present a characterisation theorem of exactly when a topological space has a ?-admissible and ?-based domain representation. We also show that there is a natural cartesian closed category of countably based and countably admissible domain representations. In Paper IV we consider admissible domain representations of convergence spaces, where a convergence space is a set X together with a convergence relation between nets on X and elements of X. We study in particular the new notion of weak ?-convergence spaces, which roughly means that the convergence relation satisfies a generalisation of the Kuratowski limit space axioms to cardinality ?. We show that the category of weak ?-convergence spaces is cartesian closed. We also show that the category of weak ?-convergence spaces that have a dense, ?-admissible, ?-continuous and ?-based consistently complete domain representation is cartesian closed when ? ? ? ? ?. As natural corollaries we obtain corresponding results for the associated category of weak convergence spaces.

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