On realizable biorders and the biorder dimension of a relation

The paper discusses the mathematical foundations of a technique of multidimensional scaling, generalizing Guttman scaling, in which the structure of the embedding space relies only on ordinal concepts. An empirical relation is represented as an intersection of a minimal number (called bidimension) of Guttman relations. Fairly complete results are given for the cases of bidimensions 1 and 2. In the general case, the main results are based on the equivalence between the bidimension and the dimension of a certain partial order. A characterization of the bidimension as the chromatic number of some hypergraph is also provided.