An algorithmic proof that semiorders are representable

doi:10.1016/0196-6774(92)90010-A

Journal of Algorithms

Volume 13, Issue 1, March 1992, Pages 144-147

https://doi.org/10.1016/0196-6774(92)90010-A Get rights and content

Abstract

For preference patterns with intransitive indifference, Scott and Suppes showed that an asymmetric digraph is representable as a consistent preference pattern if and only if it is a semiorder. Rabinovitch gave a constructive proof that every finite semiorder is representable. We give a simpler algorithmic proof of this result.

References (5)

I Rabinovitch
The Scott-Suppes theorem on semiorders
J. Math. Psych.
(1977)
P Avery
Semiorders and representable graphs

There are more references available in the full text version of this article.

Cited by (2)

Unit representation of semiorders I: Countable sets
2021, Journal of Mathematical Psychology
Citation Excerpt :
The pioneering work of Scott and Suppes (1958) on finite sets was soon followed by many other alternative proofs using various kinds of arguments. Without aiming at exhaustivity, one can cite Avery (1992), Balof and Bogart (2003), Bogart and West (1999), Isaak (2009), Rabinovitch (1977), Roberts (1971, 1979), Roubens and Vincke (1985), Roy (1996, Ch. 7), Scott (1964), Suppes and Zinnes (1963), and Troxell (2003). It is well-known that these results do not extend to the infinite case, not even to the countable case (see Fishburn, 1985 p. 30).
This paper proposes a new proof of the existence of constant threshold representations of semiorders on countably infinite sets. The construction treats each indifference-connected component of the semiorder separately. It uses a partition of such an indifference-connected component into indifference classes. Each element in the indifference-connected component is mirrored, using a “ghost” element, into a reference indifference class that is weakly ordered. A numerical representation of this weak order is used as the basis for the construction of the unit representation after an appropriate lifting operation. We apply the procedure to each indifference-connected component and assemble them adequately to obtain an overall unit representation.
Our proof technique has several original features. It uses elementary tools and can be seen as the extension of a technique designed for the finite case, using a denumerable set of inductions. Moreover, it gives us much control on the representation that is built, so that it is, for example, easy to investigate its uniqueness. Finally, we show in a companion paper that our technique can be extended to the general (uncountable) case, almost without changes, through the addition of adequate order-denseness conditions.
The parametric closure problem
2017, ACM Transactions on Algorithms

^∗: Present address: Department of Computer Science, University of California, Santa Barbara, CA 93106.

View full text

Regular articleAn algorithmic proof that semiorders are representable

Abstract

J. Math. Psych.

Semiorders and representable graphs

Regular article
An algorithmic proof that semiorders are representable