I-Fuzzy equivalence relations and I-fuzzy partitions

A T-fuzzy equivalence relation is a fuzzy binary relation on a set image which is reflexive, symmetric and T-transitive for a t-norm T. Recently, Mesiar et al. [R. Mesiar, B. Reusch, H. Thiele, Fuzzy equivalence relations and fuzzy partitions, J. Multi-Valued Logic Soft Comput. 12 (2006) 167–181] have generalised the t-norm T to any general conjunctor C and investigated the minimal assumptions required on such operations, called duality fitting conjunctors, such that the fuzzification of the equivalence relation admits any value from the unit interval and also the one–one correspondence between the fuzzy equivalence relations and fuzzy partitions is preserved. In this work, we conduct a similar study by employing a related form of C-transitivity, viz., I-transitivity, where I is an implicator. We show that although every I-fuzzy equivalence relation can be shown to be a C-fuzzy equivalence relation, there exist C-fuzzy equivalence relations that are not I-fuzzy equivalence relations and hence these concepts are not equivalent. Most importantly, we show that the class of duality fitting implicators I is much richer than the residuals of the duality fitting conjunctors in the study of Mesiar et al. We also show that the I-fuzzy partitions have a “constant-wise” structure.