On Self-Aggregations of Min-Subgroups

Abstract

Preservation of structures under aggregation functions is an active area of research with applications in many fields. Among such structures, min-subgroups play an important role, for instance, in mathematical morphology, where they can be used to model translation invariance. Aggregation of min-subgroups has only been studied for binary aggregation functions. However, results concerning preservation of the min-subgroup structure under binary aggregations do not generalize to aggregation functions with arbitrary input size since they are not associative. In this article, we prove that arbitrary self-aggregation functions preserve the min-subgroup structure. Moreover, we show that whenever the aggregation function is strictly increasing on its diagonal, a min-subgroup and its self-aggregation have the same level sets.