Commuting functions with annihilator elements

Commuting is an important property in any two-step information aggregation process where the results should not depend on the order in which the single steps are carried out. In case of aggregation functions with the neutral elements, Saminger et al. (IEEEs Transactions on Fuzzy Systems, 2007, 15, 1032–1045) already reduced characterization of commuting functions to resolving the functional equation of a unary function. In this paper, we will investigate the commuting functions with the annihilator elements, especially, the nullnorms. A characterization of the commuting aggregation functions with the annihilator elements will be obtained in this case.