Abstract
This paper gives a necessary and sufficient condition for the following proposition, in which ⪯ and ⪯i for i = 1,…, n are weak orders on a finite set X. There are real-valued functions f1, f2,…, fn on X such that, for all x and y in X and i in {1,…,n}x⪯i y if and only if fi(x) ≤ fi(y), and x ⪯ y if and only if fl(x) + ⋯ + fn(x) ≤ fl(y) + ⋯ + fn(y). The condition can be viewed as an extension of a simple unanimity or dominance condition.

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