The modularity condition for overlap and grouping functions

This paper is mainly devoted to investigating the modularity condition for overlap and grouping functions, which may have neither neutral elements nor associativity. At present, the modularity equation has been discussed in some families of certain associative operators, such as t-norms, t-conorms, uninorms and nullnorms. In this study we explore the modularity equation between overlap (grouping) functions and overlap functions, grouping functions, uninorms and nullnorms. The results that we obtained are complete and generalize the known ones about modularity for certain associative operators. (C) 2018 Elsevier B.V. All rights reserved.