We identify in this paper two conditions that characterize the domain of single-peaked preferences on the line in the following sense: a preference profile satisfies these two properties if and only if there exists a linear order $L$ over the set of alternatives such that these preferences are single-peaked with respect L. The first property states that for any subset of alternatives the set of alternatives considered as the worst by all agents cannot contains more than 2 elements. The second property states that two agents cannot disagree on the relative ranking of two alternatives with respect to a third alternative but agree on the (relative) ranking of a fourth one.