Ranking Committees, Words or Multisets

We investigate the ways in which a linear order on a finite set A can be consistently extended to a linear order on a set Pk(A) of multisets on A of fixed cardinality k. We show that for card(A) = 3 all linear orders on Pk(A) are additive and classify them by means of Farey fractions. For card(A) minor/equal 4 we show that there are non-additive consistent linear orders of Pk(A), we prove that they cannot be extended to a linear order of Pk(A) for K sufficiently large. We give the lower bounds for the number of additive linear orders in P2(A) and the total number of consistent linear orders in P2(A).