标题：The modularity condition for overlap and grouping functions
作者：Wang, Ya-Ming; Liu, Hua-Wen
作者机构：[Wang, Ya-Ming; Liu, Hua-Wen] Shandong Univ, Sch Math, Jinan 250100, Shandong, Peoples R China.
通讯作者：Liu, HW;Liu, HuaWen
通讯作者地址：[Liu, HW]Shandong Univ, Sch Math, Jinan 250100, Shandong, Peoples R China.
来源：FUZZY SETS AND SYSTEMS
关键词：Modularity; Overlap functions; Grouping functions; Nullnorms; Uninorms
摘要：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.