Crisp dimension theory and valued preference relations

Representation of binary preference relations in a real space where each coordinate suggests the existence of underlying criteria is a standard and indeed suggestive approach. Classical dimension theory addresses this problem, showing that whenever crisp preferences define a partial order set, it can be represented in a real space, and then we can search for a minimal representation. Valued preference relation being a much more complex structure, there is an absolute need for meaningful representations, being manageable by decision-makers. In this paper, we continue analyzing the concept of a generalized dimension function of valued preference relations, i.e. a mapping assigning a generalized dimension value to every α-cut of any given valued preference relation, as introduced in a previous paper. We should of course be expecting deep computational problems within this generalized dimension context, since they are already present in crisp dimension theory. In this paper, we present some properties of such a generalized dimension function, pointing out that our approach allows alternative representations depending on some underlying rationality core the decision-maker may change.