a generalization of the chermak--delgado measure on subgroups and its associated lattice

We generalize the chermak–delgado measure of a subgroup of a finite group g, µ(h) = |h||cg(h)|, and its associated lattice of subgroups with maximal measure. we consider mappings m of the lattice of all subgroups sub(g) into itself and define a measure associated to m by setting µ(h) = |h||m(h)|. we investigate under what conditions on m the subgroups with maximal measure form a sublattice of sub(g). in particular, our focus is on the case where m(h) is a centralizer-like subgroup.