FUZZY-SETS WITH INNER INTERDEPENDENCE

Fuzzy systems are fuzzy sets with interdependence between its components and/or between the propositions that describe these components or their behavior. A fuzzy system is completely characterized by a global membership function that is a probability distribution on the product space of all possible configurations of the elements of a given universe X with respect to the set of properties describing the respective system. The membership function of a standard fuzzy set A of the universe X completely defines the global membership function only in the particular case when the elements of the universe are independent with respect to the property that defines the respective fuzzy set A. The cylinder sets prove to be useful in dealing with fuzzy systems. The union, intersection, and complement of fuzzy systems may be defined in a natural way. The entropy of a fuzzy system contains as special cases the measures of uncertainty induced by fuzziness proposed by De Luca, Termini, and Hirota. The entropic measures of interdependence and total connection allow us to classify the subsystems of a fuzzy system from the point of view of their interdependence in a natural way.