Cancellation Conditions for Finite Two-Dimensional Additive Measurement

Abstract

It is well known that a weak order ≾ on a finite set X=X1×X2 has an additive real-valued order-preserving representation if and only if ≾ on X satisfies a denumerable scheme of cancellation conditions C(2), C(3), …. Condition C(K) is based on K distinct ordered pairs in X×X. Given fixed cardinalities m and n for X1 and X2, there is a largest K, denoted by f(m, n), such that some ≾ on X satisfies C(2) through C(K−1) but violates C(K). It has been known for some time that f(2, n)=2 for all n⩾2, and f(3, 3)=3. It was proved recently that f(3, n)⩾n for all even n⩾4 and that f(m, n)⩽m+n−1 for all m, n⩾2. The present paper shows that f(3, 4)=f(4, 4)=4, f(5, n)⩾n+1 for all odd n⩾5, and f(m, n)⩾m+n−10 for all odd m and n greater than or equal to 11. The last result in conjunction with the upper bound of m+n−1 shows that f(m, n) for most (m, n) is approximately m+n.