  Two distinct specifications of single peakedness as currently met in the
relevant literature are singled out and discussed. Then, it is shown that,
under both of those specifications, a voting rule as defined on a bounded
distributive lattice is strategy-proof on the set of all profiles of single
peaked total preorders if and only if it can be represented as an iterated
median of projections and constants, or equivalently as the behaviour of a
certain median tree-automaton. The equivalence of individual and coalitional
strategy-proofness that is known to hold for single peaked domains in bounded
linear orders fails in such a general setting. A related impossibility result
on anonymous coalitionally strategy-proof voting rules is also obtained.
