Some Properties of Inclusions of Multisets and Strictly Increasing Boolean Functions
Résumé
Consider the following curious puzzle: call an~$n$-tuple~$X=(X_1,...,X_n)$ of sets smaller than~$Y$ if it has less //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 strictly increasing boolean function~$f:B^n -> B^n$. We also show that contrary to plain boolean functions or increasing boolean functions, strictly increasing boolean functions aren't finitely generated, which might explain why this preorder isn't easily described concretely.
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