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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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References:
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