Some existence conditions for decomposable k-monotone set functions having no k'-monotone decompositions

Abstract

It has been already known that a set function $\mu$ defined on the power set $2^X$ of a finite set $X$ and/or the Choquet integral w.r.t. $\mu$ have a $\mathcal{C}$-decomposition iff $\mathcal{C}$ is an inclusion-exclusion covering of $X$. This paper shows that for any two positive integers $k$ and $k'$ such that $k'\le k$, there exist a finite set $X$, a $k$-monotone set function $\mu$ on $2^X$, and an inclusion-exclusion covering $\mathcal{C}$ of $X$ w.r.t. $\mu$ such that $\mu$ has no $k'$-monotone $\mathcal{C}$-decompositions.