Chain Representations of Nested Families of Biorders
Bouyssou, Denis; Doignon, Jean-Paul (2020), Chain Representations of Nested Families of Biorders, in Bosi, Gianni; Campión, María J.; Candeal, Juan C.; Indurain, Esteban, Mathematical Topics on Representations of Ordered Structures and Utility Theory : Essays in Honor of Professor Ghanshyam B. Mehta, Springer, p. 143-169. 10.1007/978-3-030-34226-5_7
Book titleMathematical Topics on Representations of Ordered Structures and Utility Theory : Essays in Honor of Professor Ghanshyam B. Mehta
Book authorBosi, Gianni; Campión, María J.; Candeal, Juan C.; Indurain, Esteban
Series titleStudies in Systems, Decision and Control (SSDC, volume 263)
Number of pages368
MetadataShow full item record
Laboratoire d'analyse et modélisation de systèmes pour l'aide à la décision [LAMSADE]
Département de Mathématique [Bruxelles] [ULB]
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.
Subjects / KeywordsBiorders; Nested relations; Interval orders; Semiorders; Numerical representations; Chain representations; Well-gradedness; Weak gradedness
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