Distributivity characterization of idempotent uni-nullnorms and overlap or grouping functions

Recently, some authors studied the distributive laws of continuous t-norms and some families of the common classes of uninorms over overlap and grouping functions in , , but until now a complete characterization of the distributivity on idempotent uninorms over overlap or grouping functions widely used in image processing is still unresolved. Moreover, authors in characterized the distributivity equations of uni-nullnorms with continuous Archimedean underlying operators over the above two functions. In this paper, we proceed with the distributivity characterization of idempotent uni-nullnorms over them which obviously generalizes the corresponding results of idempotent uninorms over these two functions. During the process, we introduce a class of weak overlap and grouping functions with weak coefficients, and obtain the full characterizations of the above functions by considering the different values of the underlying uninorms' associated functions of idempotent uni-nullnorms on the interval endpoints and comparing the values with the weak coefficients. These obtained results totally answer the question on the distributive solutions of idempotent uninorms over overlap functions, which have been mentioned as their future works in . In additions, we also obtain that overlap and grouping functions have particular structures with a constant domain where its value equals to the neutral element of the idempotent uninorm when they are distributive over idempotent uninorms.