Abstract

The linear discrepancy of a partially ordered set P=(X,≺) is the least integer k for which there exists an injection f: X→Z satisfying (i) if x≺y then f(x)<f(y) and (ii) if x∥y then |f(x)−f(y)|≤k. This concept is closely related to the weak discrepancy of P studied previously. We prove a number of properties of linear and weak discrepancies and relate them to other poset parameters. Both parameters have applications in ranking the elements of a partially ordered set so that the difference in rank of incomparable elements is minimized.

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