Failure of cancellation conditions for additive linear orders

✍ Scribed by Peter C. Fishburn

• Publisher: John Wiley and Sons
• Year: 1997
• Tongue: English
• Weight: 202 KB
• Volume: 5
• Category: Article
• ISSN: 1063-8539
• DOI: 10.1002/(sici)1520-6610(1997)5:5<353::aid-jcd4>3.0.co;2-e

✦ Synopsis

Coordinate independence assumptions, also known as cancellation conditions, play a central role in the representational theory of measurement for an ordering relation on a finite Cartesian product set A1×A2ו • •×Am. A sequence of increasingly complex cancellation conditions is known to be sufficient for additive representability in the form (a1, a2, . . . , am)

A longstanding open problem is to determine the simplest subset of cancellation conditions as a function of the size of A1 × • • • × Am that is violated by every order that is not additively representable. This article proves a lower bound on minimum subset sufficiency when all Ai are binary. We conjecture that this lower bound, which is very near to a known upper bound, is the exact minimum. The binary-factors version of the problem is reformulated under a first-order independence assumption by a map from on {0, 1} m into a subset L of {1, 0, -1} m that is referred to as an additive linear order. The lower bound is then established by examples of additive linear orders on {1, 0, -1} m that exhibit worst-case failures of cancellation.