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A representation of FS-domains by formal concept analysis. (English) Zbl 1480.18004

The category Dom of continuous domains with Scott continuous functions is not cartesian closed while from [A. Jung, Cartesian closed categories of domains. Amsterdam: Centrum voor Wiskunde en Informatica (1989; Zbl 0719.06004); “The classification of continuous domains”, in: Proceedings of the Fifth Annual IEEE Symposium on Logic in Computer Science. Los Alamitos,CA: IEEE Computer Society. 35–40 (1990)] it is known that the full subcategory FSD of FS-domains (a pointed dcpo with a directed set of continuous endomorphisms, each finitely separated from the identity morphism and having the identity morphism as its supremum [S. Abramsky and A. Jung, Domain theory, corrected and expanded version. https://www.cs.bham.ac.uk/~axj/pub/papers/handy1.pdf]) is maximal cartesian closed. The present paper proposes a notion of FS-contexts (based on the notion of contractive operators in [L. Wang et al., Fundam. Inform. 179, No. 3, 295–319 (2021; Zbl 07426112)]) and shows: each FS-domain, upto isomorphism, is the set of FS-formal concepts of a FS-context.

MSC:

18B35 Preorders, orders, domains and lattices (viewed as categories)
06B35 Continuous lattices and posets, applications
06A15 Galois correspondences, closure operators (in relation to ordered sets)
06A06 Partial orders, general
03E20 Other classical set theory (including functions, relations, and set algebra)
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