Completely lattice L-ordered sets with and without L-equality

A relationship between L-order based on an L-equality and L-order based on crisp equality is explored in detail. This enables to clarify some properties of completely lattice L-ordered sets and generalize some related assertions. Namely, Bělohlávekʹs 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 completely lattice L-ordered set of all down-L-sets is described.