Fractional weak discrepancy and split semiorders
The fractional weak discrepancy wdF(P) w d F ( P ) of a poset P=(V,?) P = ( V , ? ) was introduced in Shuchat etal. (2007) [6] as the minimum nonnegative k k for which there exists a function f:V?R f : V ? R satisfying (i) if a?b a ? b then f(a)+1?f(b) f ( a ) + 1 ? f ( b ) and (ii) if a?b a ? b then |f(a)?f(b)|?k | f ( a ) ? f ( b ) | ? k . In this paper we generalize results in Shuchat etal. (2006, 2009) [5]and[7] on the range of wdF w d F 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/s r / s for which 0?s?1?r<2s 0 ? s ? 1 ? r < 2 s .
fractional weak discrepancy split semiorders forcing cycle