Dimension of valued relations

The classical notion of dimension of a partial order can be extended to the valued setting, as was indicated in a particular case by Ovchinnikov. Relying on Valverde's result on the transitive closure of a valued relation, we define the dimension of a valued quasi order. Building then on Fodor and Roubens, we also show that the definition can be generalized to all valued relations by using valued bi-orders instead of valued weak orders as one-dimensional relations. Interesting, combinatorial questions about the new dimension concept arise and are investigated here. In particular, we aim at a characterization of valued quasi orders of dimension two.