Underlying criteria in valued preference relations

Translation of Classical Dimension Theory into a valued context should allow a comprehensible view of alternatives, by means of an informative representation, being this representation still manageable by decision makers. In fact, there is an absolute need for this kind of representations, since being able to comprehend a valued preference relation is most of the time the very first difficulty decision makers afford, even when dealing with a small number of alternatives. Moreover, we should be expecting deep computational problems, already present in classical crisp Dimension Theory. A natural approach could be to analyze dimension of every α-cut of a given valued preference relation. But due to complexity in dealing with valued preference relations, imposing max-min transitivity to decision makers in order to assure that every α-cut defines a crisp partial order set seems quite unrealistic. In this paper we propose an alternative definition of crisp dimension, based upon a general representation result, that may allow the possibility of skipping some of those computational problems.