Generalized semimodularity: order statistics
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A notion of generalized n-semimodularity is introduced, which extends that of (sub/super)modularity in four ways at once. The main result of this paper, stating that every generalized (n2)-semimodular function on the nth Cartesian power of a distributive lattice is generalized n-semimodular, may be considered a multi/infinite-dimensional analogue of the well-known Muirhead lemma in the theory of Schur majorization. This result is also similar to a discretized version of the well-known theorem due to Lorentz, which latter was given only for additive-type functions. Illustrations of our main result are presented for counts of combinations of faces of a polytope; one-sided potentials; multiadditive forms, including multilinear ones -- in particular, permanents of rectangular matrices and elementary symmetric functions; and association inequalities for order statistics. Based on an extension of the FKG inequality due to Rinott & Saks and Aharoni & Keich, applications to correlation inequalities for order statistics are given as well.
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