  In this article, we study exponents which preserve complete monotonicity of
functions on lattices. We prove that for any completely monotone function $f$
on a finite lattice, $f^\alpha$ is completely monotone for all $\alpha\geq c$,
where $c$ is explicitly described. For finite distributive lattices we show
that the bound $c$ is sharp. Important examples of completely monotone
functions are void functionals of random closed sets. We prove that if
$V_{\mathcal{X}}$ is the void functional of a random subset $\mathcal{X}$ of
$[n]$, then $V_{\mathcal{X}}^\alpha$ is void functional of some random closed
set for $\alpha\geq n-1$. The results are analogous to the result of FitzGerald
and Horn on Hadamard powers of positive semi-definite matrices. Also, we study
the question of approximating an $m$-divisible random set by infinitely
divisible random sets, and its generalization to lattices.
