 Other domain-restriction conditions with similar implications include single-cavedness, a geometrical mirror image of single-peakedness (Inada 1964), separability into two groups (ibid.), and latin-squarelessness (Ward 1965), the latter two more complicated combinatorial conditions (for a review, see Gaertner 2001). Sen (1966) showed that all these conditions imply a weaker condition, triple-wise value-restriction. It requires that, for every triple of alternatives \(x, y, z \in X\), there exists one alternative in \(\{x, y, z\}\) and one rank \(r \in \{1, 2, 3\}\) such that no individual ranks that alternative in \(r\)th place among \(x, y\), and \(z\). For instance, all individuals may agree that \(y\) is not bottom \((r = 3)\) among \(x, y\), and \(z\). Triple-wise value-restriction suffices for transitive majority preferences.