Abstract (EN)

Among the real-valued representations of nested families of biorders some representations reflect the nestedness of the family in a simple way. Calling them chain representations, we prove their existence in the finite and countably infinite cases. For the general case, we obtain chain representations in a well-chosen linearly ordered set. Although the existence of real-valued representations in general remains an open problem, our analysis answers questions left pending in the literature. It also leads to new proofs of classical theorems on the existence of a real representation for a single biorder, as well as for a single interval order. A combinatorial property of the set of all biorders from a finite set to another finite set plays a central role in the new proof; called weak gradedness, it is a particularization of well-gradedness which derives from a simpler argument.