Restricted equivalence functions induced from fuzzy implication functions

Restricted equivalence function, as an effective tool for the theoretical research and practical applications of fuzzy sets and systems along with fuzzy logic, has been continuously considered by scholars since it was proposed. In particular, recently, Bustince, Campi´on, De Miguel et al. (H. Bustince, M.J. Campi´on, L. De Miguel, E. Indur´ain, Strong negations and restricted equivalence functions revisited: An analytical and topological approach, Fuzzy Sets and Systems (2021), https://doi.org/10.1016/j.fss.2021.10.013.) investigated it using analytical and topological approach and proposed an open problem to ask whether the binary function F (x, y) = T (I(x, y), I(y, x)) obtained from a t-norm T and a fuzzy implication function I is a restricted equivalence function or not. In this paper, we pay attention to this problem and give positive answer of it. Specifically, first, we consider the binary functions obtained from overlap functions and fuzzy implication functions by following the construction way of F and get the necessary and sufficient condition that makes such obtained F to be a restricted equivalence function. Second, we introduce the so-called -functions, which are binary functions on unit closed interval with few additional axioms and obtain the necessary and sufficient condition that ensures the binary function constructed via any non-decreasing -function and fuzzy implication function as the way of F to be a restricted equivalence function. Finally, we give the necessary and sufficient condition that makes F to be a restricted equivalence function.