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A Generalization of Teo and Sethuraman's Median Stable Marriage Theorem

Let $L$ be any finite distributive lattice and $B$ be any boolean predicate defined on $L$ such that the set of elements satisfying $B$ is a sublattice of $L$. Consider any subset $M$ of $L$ of size $k$ of elements of $L$ that satisfy $B$. Then, we show that $k$ generalized median elements generated from $M$ also satisfy $B$. We call this result generalized median theorem on finite distributive lattices. When this result is applied to the stable matching, we get Teo and Sethuraman's median stable matching theorem. Our proof is much simpler than that of Teo and Sethuraman. When the generalized median theorem is applied to the assignment problem, we get an analogous result for market clearing price vectors.