The Submodular Inequality of Aggregation Operators

Aggregation operators have become an essential tool in many applications. The functional equations related to aggregation operators play an important role in fuzzy sets and fuzzy logic theory. The modular equation is strongly connected with the distributivity equation and can be considered as a constrained associative equation. In this paper, we consider the submodular inequality, which can be viewed as a generalization of the modular equation. First, we discuss the submodular inequality of two general aggregation operators under duality and isomorphism. Moreover, one result of the submodular inequality is presented for the ordinal sum aggregation operators. In the cases of triangular norms and triangular conorms, we present the solutions and validate the symmetry in the related results for some classes of aggregation operators.