Lattice type fuzzy order and closure operators in fuzzy ordered sets

Complete lattices and closure operators in ordered sets are considered from the point of view of fuzzy logic. A typical example of a fuzzy order is the graded subsethood of fuzzy sets. Graded subsethood makes the set of all fuzzy sets in a given universe into a completely lattice fuzzy ordered set (i.e. a complete lattice in fuzzy setting). Another example of a completely lattice fuzzy ordered set is the set of all so-called fuzzy concepts in a given fuzzy context; the respective fuzzy order is the graded subconcept/superconcept relation. Conversely, each completely lattice fuzzy ordered set is isomorphic to some fuzzy ordered set of fuzzy concepts of a given fuzzy context. These natural examples motivate us to investigate some general properties of complete lattice-type fuzzy order. Particularly, the article focuses mainly on closure operators in fuzzy ordered sets.