Cancellation Conditions for Finite Two-Dimensional Additive Measurement.
It is well known that a weak order similar on a finite set X=X(1)xX(2) has an additive real-valued order-preserving representation if and only if similar on X satisfies a denumerable scheme of cancellation conditions C(2), C(3), em leader. Condition C(K) is based on K distinct ordered pairs in XxX. Given fixed cardinalities m and n for X(1) and X(2), there is a largest K, denoted by f(m, n), such that some similar 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. Copyright 2001 Academic Press.
Fishburn
Peter C.
PC
AT & T Labs-Research