Predicative theories of continuous lattices
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Publication:6342520
DOI10.23638/LMCS-17(2:22)2021arXiv2006.05642MaRDI QIDQ6342520
Publication date: 9 June 2020
Abstract: We introduce a notion of strong proximity join-semilattice, a predicative notion of continuous lattice which arises as the Karoubi envelop of the category of algebraic lattices. Strong proximity join-semilattices can be characterised by the coalgebras of the lower powerlocale on the wider category of proximity posets (also known as abstract bases or R-structures). Moreover, locally compact locales can be characterised in terms of strong proximity join-semilattices by the coalgebras of the double powerlocale on the category of proximity posets. We also provide more logical characterisation of a strong proximity join-semilattice, called a strong continuous finitary cover, which uses an entailment relation to present the underlying join-semilattice. We show that this structure naturally corresponds to the notion of continuous lattice in the predicative point-free topology. Our result makes the predicative and finitary aspect of the notion of continuous lattice in point-free topology more explicit.
Topological spaces and generalizations (closure spaces, etc.) (54A05) Frames, locales (06D22) Continuous lattices and posets, applications (06B35) Frames and locales, pointfree topology, Stone duality (18F70)
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