A characterization of superdictatorial domains for strategy-proof social choice functions

A domain D is dictatorial iff there exists no surjective, strategy-proof and non-dictatorial social choice function defined over D. A dictatorial domain D is superdictatorial iff every superdomain of D is also dictatorial. The existence of dictatorial but not superdictatorial domains being known, we show that a dictatorial domain D is superdictatorial iff every alternative is ranked at the top by some ordering in D.