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Biorder families, valued relations, and preference modelling. (English) Zbl 0612.92020

Some results contributed to the theory of linear structures are presented. The basic concept in these results is the biorder [see the first author, A. Ducamp and J. C. Falmagne, ibid. 28, 73-109 (1984; Zbl 0562.92018)]. Several definitions of biorder are listed, useful for applications as numerical and matricial representations, or for their easy testability. Considering arbitrary families of relations the equivalence of different consistency conditions is proved, which involve results considered by F. S. Roberts [see e.g. ibid. 8, 248- 263 (1971; Zbl 0223.92017)] and P. C. Fishburn [see ibid. 7, 144- 149 (1970; Zbl 0191.315)].
Various new concepts of left- (or right-) homogeneous families of biorders, interval orders and interval duorders are formulated. For the case when irreflexivity is required the concepts of homogeneous families of coherent biorders, semiorders, and semiduorders are introduced.
Basing on the given results, left- (right-) linearly biordered, interval and coherently biordered, and semiordered valued relations and probabilistic valued relations are considered. Fields of applications for the presented results and connections with existing work are mentioned.
Reviewer: F.Burshtein

MSC:

91E99 Mathematical psychology
06A06 Partial orders, general
91B08 Individual preferences
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