Some properties of inclusions of multisets and contractive boolean operators

Consider the following curious puzzle: call an  n -tuple  X ¯ = ( X 1 , … , X n ) of sets smaller than another  n -tuple  Y ¯ if it has fewer unordered sections. We show that equivalence classes for this preorder are very easy to describe and characterize the preorder in terms of the simpler pointwise inclusion and the existence of a special increasing boolean operator  f : B n → B n . We also show that contrary to increasing boolean operators, the relevant operators are not finitely generated, which might explain why this preorder is not easy to describe concretely.