Failure of cancellation conditions for additive linear orders

Abstract

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 × A2 × ··· × Am. A sequence of increasingly complex cancellation conditions is known to be sufficient for additive representability in the form (a1, a2, ⋖ am) ≻ (b1, b2, ⋖ bm) ⇔ ∑i v(ai) > &sumi v(bi). 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. © 1997 John Wiley & Sons, Inc. J Combin Designs 5:353–365, 1997