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Completely lattice \(L\)-ordered sets with and without \(L\)-equality. (English) Zbl 1218.06003
Summary: A relationship between \(L\)-order based on an \(L\)-equality and \(L\)-order based on crisp equality is explored in detail. This enables us to clarify some properties of completely lattice \(L\)-ordered sets and generalize some related assertions. Namely, Bělohlávek’s main theorem of fuzzy concept lattices is generalized as well as his theorem dealing with Dedekind-MacNeille completion. Analogously, completion of an \(L\)-ordered set via the completely lattice \(L\)-ordered set of all down-\(L\)-sets is described.

MSC:
06A75 Generalizations of ordered sets
06B75 Generalizations of lattices
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