The Chermak–Delgado measure as a map on posets

The Chermak–Delgado measure of a finite group is a function which assigns to each subgroup a positive integer. In this paper, we give necessary and sufficient conditions for when the Chermak–Delgado measure of a group is actually a map of posets, i.e., a monotone function from the subgroup lattice to the positive integers. We also investigate when the Chermak–Delgado measure, restricted to the centralizers, is increasing.