Lattice-type fuzzy order is uniquely given by its 1-cut: proof and consequences

A 1-cut of a fuzzy relation (sometimes called a core) does not contain all the information that is represented by the fuzzy relation. Particularly, a fuzzy order <= on a universe X equipped with an fuzzy equality (almost equal) is not uniquely determined by its 1-cut 1<= ={(left angle bracket) x,y (right-pointing angle bracket) | (x<=y)=1}. That is, there are in general several fuzzy orders with a common 1-cut. We show that, if the fuzzy order obeys in addition the lattice structure (which many natural examples of fuzzy orders do), it is uniquely determined by its 1-cut. Moreover, we discuss consequences of this result for the so-called fuzzy concept lattices and formal concept analysis.