We consider the lattice, , of all subsets of a multidimensional contingency table and establish the properties of monotonicity and supermodularity for the marginalization function, , on . We derive from the supermodularity of some generalized Fr\'echet inequalities complementing and extending inequalities of Dobra and Fienberg. Further, we construct new monotonic and supermodular functions from , 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.
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