Some properties of inclusions of multisets and contractive boolean operators

Consider the following curious puzzle: call an  nn-tuple  X¯=(X1,…,Xn) of sets smaller than another  nn-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:Bn→Bn. 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.