In 1978, Rudolf Ahlswede and David Daykin published a theorem which says that a certain inequality on nonnegative real valued functions for pairs of points in a finite distributive lattice extends additively to pairs of lattice subsets. It is an elegant theorem with widespread applications to inequalities for systems of subsets, linear extensions of partially ordered sets, and probabilistic correlation. We review the theorem and its applications, and describe a recent generalization to n-tuples of points and subsets in distributive lattices. Although many implications of the Ahlswede-Daykin theorem follow from the weaker hypotheses of the widely-cited FKG theorem, several important implications are noted to require the stronger hypotheses of the basic theorem of Ahlswede and Daykin.
KeywordsPartial Order Distributive Lattice Random Permutation Linear Extension Strong Hypothesis
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