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 (1984) (Ovchinnikov, S.V. 1984. Representations of transitive fuzzy relations. In: Skala, H.J. Termini, S. Trillas, E. (Eds.), Aspects of vagueness. Reidel, Boston, pp. 105-118). Relying on Valvede's result (1985) (Valverde, L. 1985. On the structure of F-indistinguishability operators. Fuzzy Sets and Systems 17, 313-328) on the transitive closure of a valued relation, we define the dimension of a valued quasi order. Building then on Fodor and Roubens (1995) (Fodor, J. Roubens M. 1995. Structure of transitive valued binary relations. Mathematical Social Sciences, 30, 71-94), we also show that the definition can be generalized to all valued relations by using valued biorders 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.SCOPUS: ar.j