Fractional weak discrepancy and split semiorders

The fractional weak discrepancy  wdF(P)wdF(P) of a poset P=(V,≺)P=(V,≺) was introduced in Shuchat et al. (2007) [6] as the minimum nonnegative kk for which there exists a function f:V→Rf:V→R satisfying (i) if a≺ba≺b then f(a)+1≤f(b)f(a)+1≤f(b) and (ii) if a∥ba∥b then |f(a)−f(b)|≤k|f(a)−f(b)|≤k. In this paper we generalize results in Shuchat et al. (2006, 2009) [5] and [7] on the range of wdFwdF for semiorders to the larger class of split semiorders. In particular, we prove that for such posets the range is the set of rationals that can be represented as r/sr/s for which 0≤s−1≤r<2s0≤s−1≤r<2s.