Uniform fuzzy relations and fuzzy functions
In this paper, one of the approaches to the concept of a fuzzy function is presented. Diverse types of fuzzy functions are based on fuzzy equivalence relations. For example, such an approach was used in definitions of partial fuzzy functions and fuzzy functions presented by Klawonn, and of strong fuzzy functions and perfect fuzzy functions presented by Demirci. In this paper, a new approach, different from the ones mentioned above, is given. The authors present characterizations of partial and perfect fuzzy functions. They also define uniform fuzzy relations and give various characterizations of such relations. Some necessary and sufficient conditions for the existence of a uniform fuzzy relation of \(A\) to \(B\) which induces a given pair of fuzzy equivalences \(E\) on \(A\) and \(F\) on \(B\) are presented. One of them is the existence of a bijective function of the quotient set \(A/E\) onto the quotient set \(B/F\). Moreover, the concept of an \(F\)-function is developed. A partially uniform \(F\)-function is defined here as an \(F\)-function which is a partially uniform fuzzy relation, and a perfect \(F\)-function is defined as an \(F\)-function which is a perfect fuzzy function. Necessary and sufficient conditions for the existence of a uniform \(F\)-function from \(A\) to \(B\) which induces a given fuzzy equivalence on \(A\) and a given fuzzy equality on \(B\) are determined. Furthermore, two constructions of uniform \(F\)-functions are presented. Since relational composition of two partially uniform \(F\)-functions is not necessarily a partially uniform \(F\)-function, another kind of composition, which preserves the property to be a uniform \(F\)-function and the property to be a perfect \(F\)-function, is introduced. Except for the mentioned two types of compositions, a third one is also introduced and relationships between these three kinds of compositions are studied. The definition of a uniform \(F\)-function presented in this paper enables a mutual correspondence between fuzzy functions and fuzzy equivalences, analogous to the correspondence between crisp functions and crisp equivalences. In 1986, Nemitz also tried to give such a correspondence, but his approach and results are not the same. The similarities and differences between Nemitz's fuzzy functions and uniform \(F\)-functions considered by the authors are also presented in the paper. Finally, some applications of uniform fuzzy relations in approximate reasoning, especially in fuzzy control, are discussed. It is shown that uniform fuzzy relations are closely related to the defuzzification problem. The presented problem deals with a control situation when the input/output values are represented by two fuzzy partitions of the input/output universe of discourse. A system of fuzzy relation equations determined by these fuzzy partitions is considered. A necessary and sufficient condition for solvability of this system is given. It is also shown that under these conditions the solution can be constructed as a uniform fuzzy relation.
scientific article; zbMATH DE number 5634424