ORDERING FUZZY SETS BY CONSENSUS CONCEPT AND FUZZY RELATION EQUATIONS

A general class of consensus measures of fuzzy sets is introduced in this paper. It is shown that, while the consensus measures are valuations but neither isotone nor antitone with respect to the lattice structure induced by the pointwise maximum and minimum operations to the set of all the fuzzy sets on a nonempty crisp set, they are antitone valuations with respect to the lattice structure induced by the generalized sharpening relation to a quotient set of the set of fuzzy sets. It is also shown that the solutions of a finite fuzzy relation equation that have the maximum consensus measure can be characterized through the join operation of the latter lattice in terms of the maximum solution and some of the minimal solutions of the equation.