Meet-continuity and locally compact sober dcpos

Jia, Xiaodong (2018). Meet-continuity and locally compact sober dcpos. University of Birmingham. Ph.D.

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Abstract

In this thesis, we investigate meet-continuity over dcpos. We give different equivalent descriptions of meet-continuous dcpos, among which an important characterisation is given via forbidden substructures. By checking the function space of such substructures we prove, as a central contribution, that any dcpo with a core-compact function space must be meet-continuous. As an application , this result entails that any cartesian closed full subcategory of quasicontinuous domains consists of continuous domains entirely. That is to say , both the category of continuous domains and that of quasicontinuous domains share the same cartesian closed full subcategories.
Our new characterisation of meet-continuous dcpos also allows us to say more about full subcategories of locally compact sober dcpos which are generalisations of quasicontinuous domains. After developing some theory of characterising coherence and bicompleteness of dcpos, we conclude that any cartesian closed full subcategory of pointed locally compact sober dcpos is entirely contained in the category of stably compact dcpos or that of L-dcpos.
As a by-product, our study of coherence of dcpos enables us to characterise Lawson-compactness over arbitrary dcpos.

Type of Work:

Thesis (Doctorates > Ph.D.)

Award Type:

Doctorates > Ph.D.

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