Existence of homogeneous representations of interval orders on a cone in a topological vector space

Necessary and sufficient conditions are presented for the existence of a pair of positively homogeneous of degree one real functions representing an interval order < on a real cone K in a topological vector space E (in the sense that, for every x,y in K, x < y if and only if v(x) < u(y), with u lower semicontinuous, v upper semicontinuous, and u and v utilityfunctions for two complete preorders intimately connected with <. We conclude presenting a new approach to get such kind of representations, based on the concept of a biorder