Consider two elements formula_79 and formula_26 of formula_39. We are mainly interested in the case where formula_79 and formula_26 are integral, but this assumption is not necessary to the definition we are about to introduce. We then say that formula_79 is higher than formula_26, which we write as formula_86, if formula_87 is expressible as a linear combination of positive roots with non-negative real coefficients. This means, roughly, that "higher" means in the directions of the positive roots. We equivalently say that formula_26 is "lower" than formula_79, which we write as formula_90.