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.