New results on overlap and grouping functions

Overlap functions and grouping functions are special kinds of aggregation operators that have been recently proposed for applications in classification problems, like, e.g., imaging processing. Overlap and grouping functions can also be applied in decision making based on fuzzy preference relations, where the associativity property is not strongly required and the use oft-norms or t-conorms as the combination/separation operators is not necessary. The concepts of indifference and incomparability defined in terms of overlap and grouping functions may allow the application in several different contexts. This paper introduces new interesting results related to overlap and grouping functions, investigating important properties, such as migrativity, homogeneity, idempotency and the existence of generators. De Morgan triples are introduced in order to study the relationship between those dual concepts. In particular, we introduce important results related to the action of automorphisms on overlap and grouping functions, analyzing the preservation of those properties and also the Lipschitzianity condition. (C) 2013 Elsevier Inc. All rights reserved.