Arrow incorporated into the notion of a social welfare function the further requirement that \(f\langle R_{i}\rangle\) is always a weak ordering of the set \(X\) of alternatives.  Informally speaking, this means that the output of the social welfare function must always be a ranking of the alternatives from better to worse, perhaps with ties. It may never be a cycle. This requirement will appear here, as it does in other contemporary presentations of Arrow’s theorem, as a separate condition of Social Ordering that social welfare functions might be required to meet.  See  Section 3.1. This way, we can consider the  consequences of dropping this condition without changing any basic parts of the framework. See Section 4.2.