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Linear finitely separated objects of subcategories of domains. (English) Zbl 0890.06007
The category $${\mathcal F}rm$$ of frames and frame homomorphisms with the important concept of approximation by specifying a full subcategory of linear FS-frames is enriched. It is shown that this subcategory is equivalent with the subcategory of linear FS-domains via the standard Stone duality. The Stone duality for sober spaces gives that a distributive continuous lattice, i.e., a continuous frame can be viewed as the lattice of open sets of a locally compact space. The Stone duals of linear FS-domains considered as topological spaces, LFS-frames, may be replaced by their suitably taken distributive sublattices with an additional relation of approximation, thus discarding with infinitary operations. This is intended as a step towards the development of a domain theory in logical form beyond the standard algebraic world. Moreover, since any linear FS-frame is stably continuous and supercontinuous, it is possible to characterize the full category of stably continuous supercontinuous frames to be equivalent to the subcategory of stable prelocales (distributive lattices with an approximation relation and stable approximable relations between them).
##### MSC:
 06D20 Heyting algebras (lattice-theoretic aspects)
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