[This paper simplifies and extends some previous work on domain restriction. With the help of an easy arithmetical result, two types of necessary and sufficient conditions are established for transitivity and quasi-transitivity under majority rule and transitivity under non-minority rule.]

[A social decision rule is one that produces a social decision for each configuration of individuals' decisions. Such a rule is representative if it produces a social decision that is the result of repeatedly applying the rule of simple majority decision to decisions obtained by that rule. We give necessary and sufficient conditions for a social decision rule for two alternatives to be representative.]