Fuzzy Equivalence Relations and Fuzzy Partitions
R. Mesiar, B. Reusch and H. Thiele
Several ways of fuzzifying classical equivalence relations and partitions were introduced and discussed in the literature, mostly based on fuzzy logic conjunctor modeled by a t-norm. We discuss the weakest properties of such operators allowing to introduce fuzzy equivalence relations and fuzzy partitions. First, the fuzzification admitting all values from the unit interval is discussed. Fuzzifications based on conjunctors should be a proper extension of classical equivalence relations and partitions. The one–to–one correspondence between crisp equivalence relations and partitions should have a counter-part in fuzzy framework; this point of view leads to the same class of fitting conjunctors for fuzzy equivalence relations and fuzzy partitions. The relationship to triangular norms is shown, too. Several examples are introduced. The structure of fitting conjunctors based fuzzy equivalence relations and fuzzy partitions is investigated. In the case of generated conjunctors, the relationship of fuzzy equivalence relations and pseudo-metrics is shown. Finally, the aggregation of fuzzy equivalence relations, as well as fuzzy partitions, is discussed.