PREDICATIVE THEORIES OF CONTINUOUS LATTICES

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.

Automated summary

The notion of strong proximity join-semilattice is introduced as a predicative notion of continuous lattice, characterized by coalgebras of the powerlocale on proximity posets. Locally compact locales can also be characterized in terms of strong proximity join-semilattices. A logical characterization of a strong proximity join-semilattice, known as a strong continuous finitary cover, using an entailment relation is also provided, making the notion of continuous lattice in point-free topology more explicit.