  We consider the lattice, $\mathcal{L}$, of all subsets of a multidimensional
contingency table and establish the properties of monotonicity and
supermodularity for the marginalization function, $n(\cdot)$, on $\mathcal{L}$.
We derive from the supermodularity of $n(\cdot)$ some generalized Fr\'echet
inequalities complementing and extending inequalities of Dobra and Fienberg.
Further, we construct new monotonic and supermodular functions from $n(\cdot)$,
and we remark on the connection between supermodularity and some correlation
inequalities for probability distributions on lattices. We also apply an
inequality of Ky Fan to derive a new approach to Fr\'echet inequalities for
multidimensional contingency tables.
