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Admissible domain representations of convergence spaces
Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics.
Manuscript (Other academic)
Identifiers
URN: urn:nbn:se:uu:diva-93283OAI: oai:DiVA.org:uu-93283DiVA: diva2:166718
Available from: 2005-09-07 Created: 2005-09-07 Last updated: 2010-01-13Bibliographically approved
In thesis
1. Effective Domains and Admissible Domain Representations
Open this publication in new window or tab >>Effective Domains and Admissible Domain Representations
2005 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

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 λ-admissible if, in principle, all other λ-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 λ-admissible and κ-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 κ-convergence spaces, which roughly means that the convergence relation satisfies a generalisation of the Kuratowski limit space axioms to cardinality κ. We show that the category of weak κ-convergence spaces is cartesian closed. We also show that the category of weak κ-convergence spaces that have a dense, λ-admissible, κ-continuous and α-based consistently complete domain representation is cartesian closed when α ≤ λ ≥ κ. As natural corollaries we obtain corresponding results for the associated category of weak convergence spaces.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2005. vii + 33 p.
Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 42
Keyword
Logic, symbolic and mathematical, domain theory, admissible domain representation, cartesian closure, effective domains, κ-sequential space, limit space, Matematisk logik
National Category
Algebra and Logic
Identifiers
urn:nbn:se:uu:diva-5883 (URN)91-506-1817-2 (ISBN)
Public defence
2005-09-29, Room 1211, Building 1, Polacksbacken, Uppsala, 10:15
Opponent
Supervisors
Available from: 2005-09-07 Created: 2005-09-07Bibliographically approved

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