Basic theorem of fuzzy concept lattices revisited

There are two versions of the basic theorem of L-concept lattices for L being a complete residuated lattice, both proved by Belohlavek: the crisp order version and the fuzzy order version. We introduce a third version, equivalent to the fuzzy order version, but simpler and related more closely to the classical Wille's basic theorem of concept lattices. Then we use it to prove some new results on substructures of L-concept lattices and show a simpler proof of a known result on factor structures of L-concept lattices. We show by means of several counterexamples that the crisp order version does not describe the structure of L-concept lattices sufficiently. We argue 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. We also discuss the correspondence between the Belohlavek's fuzzy order version of the basic theorem and the version introduced in this paper.