On the existence of an arrow and a Bergson-Samuelson social welfare function

Let Σ be the set of all possible preferences over a given set of alternatives A. Let Ω be a proper subset of Σ and let Pε{lunate}Ωⁿ be a fixed profile of preferences. P is heterogeneous in Ω if for all a,b,cε{lunate}A and Qε{lunate}Ωⁿ, there exist three alternatives x,y,zε{lunate}A such that Q(a,b,c)=P(x,y,z) where Q(B) denotes the subprofile over a set of alternatives B⊂A. An Arrow SWF f{hook} is a function f{hook}:Ωⁿ→Σ satisfying the conditions Pareto and IIA. A Bergson-Samuelson SWF is a function f{hook}:P→Σ satisfying Pareto and Independence+Neutrality. The paper shows that (a) there exist a neutral nondictatorial Arrow SWF on Ω if and only if there exist a neutral nondictatorial Bergson-Samuelson SWF on P. (b) There exist a nondictatorial n person Bergson-Samuelson SWF on P if and only if there exists a 3 person Bergson-Samuelson SWF on P. (c) There exists a nondictatorial Arrow SWF on Ω if and only if there exists a nondictatorial Bergson-Samuelson SWF on P.