The modularity condition for uninorms revisited.

Functional equations involving aggregation functions play an important role in fuzzy sets and fuzzy logic theory. The modularity equation is a kind of restricted general associative equations. In the literature, the already known results concerning the modularity equation for two uninorms are based on the assumption that both uninorms lie in one of the most studied classes of uninorms. In this study we explore the modularity equation involving uninorms in a most general setting. Specifically, we investigate the modularity between two uninorms in the cases when one uninorm belongs to any of the most studied classes of uninorms, but the other one is any uninorm with no further assumptions. Further, many new solutions appear from this new point of view that were not included in previous approaches.