Removal independent consensus methods for closed [beta]-systems of sets

Let [beta] be a positive integer and let E be a finite nonempty set. A closed [beta]-system of sets on E is a collection H of subsets of E such that A[set membership, variant]H implies A>=[beta], E[set membership, variant]H, and A[intersection]B[set membership, variant]H whenever A,B[set membership, variant]H with A[intersection]B>=[beta]. If is a class of closed [beta]-systems of sets and n is a positive integer, then is a consensus method. In this paper we study consensus methods that satisfy a structure preserving condition called removal independence. The basic idea behind removal independence is that if two input profiles P,P* in agree when restricted to a subset A of E, then their consensus outputs C(P),C(P*) agree when restricted to A. By working with the axiom of removal independence and classes of closed [beta]-systems of sets we obtain a result for consensus methods that is in the same spirit as Arrow's Impossibility Theorem for social welfare functions.