Equivalent bipolar fuzzy relations

We consider a relation between bipolar fuzzy relations (originally called intuitionistic fuzzy relations). This relation turns out to be an equivalence in the family of all bipolar fuzzy relations in a given (crisp) set. It also has many other properties which seem to be useful in applications. We study standard properties of bipolar fuzzy relations in the context of the considered equivalence relation. Moreover, we propose new types of properties for bipolar fuzzy relations which are “compatible” with the regarded relation and we put some connections between these new types of properties and the standard ones.