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Basic theorem of fuzzy concept lattices revisited. (English) Zbl 1385.06006

Authors’ abstract: There are two versions of the basic theorem of \(L\)-concept lattices for \(L\) being a complete residuated lattice, both proved by R. Bělohlávek: the crisp-order version [Math. Log. Q. 47, No. 1, 111–116 (2001; Zbl 0976.03025)] and the fuzzy-order version [Ann. Pure Appl. Logic 128, No. 1–3, 277–298 (2004; Zbl 1060.03040)]. In this paper, the author introduces a third version, equivalent to the fuzzy-order version, but simpler and related more closely to the classical basic theorem of concept lattices by R. Wille [in: Ordered sets. Proceedings of the NATO Advanced Study Institute held at Banff, Canada, 1981. Dordrecht-Boston-London: D. Reidel Publishing Company. 445–470 (1982; Zbl 0491.06008)]. Then, he uses it to prove some new results on substructures of \(L\)-concept lattices and shows a simpler proof of a known result on factor structures of \(L\)-concept lattices. He shows by means of several counterexamples that the crisp order version does not describe the structure of \(L\)-concept lattices sufficiently. He argues that in order to formulate and prove theoretical results on \(L\)-concept lattices that are similar to those known from classical formal concept analysis, it is essential to use the fuzzy order version of the basic theorem. He also discusses the correspondence between Belohlavek’s [2004, loc. cit.] fuzzy-order version of the basic theorem and the version introduced in this paper.

MSC:

06D72 Fuzzy lattices (soft algebras) and related topics
06A15 Galois correspondences, closure operators (in relation to ordered sets)
06B23 Complete lattices, completions
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