Factor structures and central points by similarity

We present a general framework and results on factorization of systems of fuzzy sets by similarity. The result of such factorization can be regarded as a simplified version of the original system in which we deliberately do not distinguish elements which are highly similar. We assume that the fuzzy sets are fixed points of some closure operator. Examples of such systems are fuzzy concept lattices, fuzzy sets in a given universe, and complete residuated lattices. The similarity relation we consider is an a-cut of a particular fuzzy equivalence relation with a being a similarity threshold supplied by a user which controls the meaning of ldquohighly similarrdquo. We present results describing the factorization including an efficient way to compute the factor structure. In addition, we describe a-central points of a given collection of fixed points of a closure operator, i.e. points which are similar to every point in the collection to degree at least a.