A note on the Suzumura-consistency

In Suzumura [Suzumura K., 1976. Remarks on the theory of collective choice. Economica 43, 381-90], it is proved that a binary relation has an ordering extension if and only if it is Suzumura-consistent. We show necessary and sufficient conditions for a binary relation to have ordering extensions rationalizing a unique set of alternatives and further, for this set of alternatives to be a singleton.