Dimension of valued relations

Abstract

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 Valverde'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.