A characterization of the single-peaked domain

We provide in this paper a pair of properties that are both necessary and sufficient to characterize the domain of single-peaked preference profiles. This characterization allows for a definition of single-peaked preference profiles without using an ad-hoc unlerlying order of the alternatives and also sheds light on the structure of single-peaked profiles. Considering the larger domain of value-restricted preference profiles (Sen, A Possibility Theorem on Majority Decisions, Econometrica, 1966) we also provide necessary and sufficient conditions for a preference profile to be single-caved or group-separable. Our results show that for single-peaked, single-caved and group-separable profiles it is sufficient to restrict to profiles containing of either three individuals and three alternatives or two individuals and four alternatives.