Diestel, et al. (see Order 35 (2017), JCT-A 167 (2019), arXiv:1805.01439) introduced the notion of abstract separation systems that satisfy a submodularity property, and they call this structural submodularity. Williamson, Goemans, Mihail, and Vazirani (Combinatorica 15 (1995)) call a family of sets $\mathcal{F}$ uncrossable if the following holds: for any pair of sets $A,B\in\mathcal{F}$, both $A\cap{B},A\cup{B}$ are in $\mathcal{F}$, or both $A-B,B-A$ are in $\mathcal{F}$. Bansal, Cheriyan, Grout, and Ibrahimpur (Algorithmica 86 (2024), arXiv:2209.11209) call a family of sets $\mathcal{F}$ pliable if the following holds: for any pair of sets $A,B\in\mathcal{F}$, at least two of the sets $A\cap{B},A\cup{B},A-B,B-A$ are in $\mathcal{F}$. We say that a pliable family of sets $\mathcal{F}$ satisfies structural submodularity if the following holds: for any pair of crossing sets $A,B\in\mathcal{F}$, at least one of the sets $A\cap{B},A\cup{B}$ is in $\mathcal{F}$, and at least one of the sets $A-B,B-A$ is in $\mathcal{F}$. For any positive integer $d\geq2$, we construct a pliable family of sets $\mathcal{F}$ that satisfies structural submodularity such that (a) there do not exist a symmetric submodular function $g$ and $λ\in{\mathbb Q}$ such that $\mathcal{F} = \{ S \,:\, g(S)<λ\}$, and (b) $\mathcal{F}$ cannot be partitioned into $d$ (or fewer) uncrossable families.

翻译：Diestel 等人（参见 Order 35 (2017), JCT-A 167 (2019), arXiv:1805.01439）引入了满足次模性性质的抽象分离系统概念，并将其称为结构次模性。Williamson、Goemans、Mihail 和 Vazirani (Combinatorica 15 (1995)) 称一个集合族 $\mathcal{F}$ 为不可交叉的，如果满足：对于任意集合对 $A,B\in\mathcal{F}$，要么 $A\cap{B},A\cup{B}$ 都属于 $\mathcal{F}$，要么 $A-B,B-A$ 都属于 $\mathcal{F}$。Bansal、Cheriyan、Grout 和 Ibrahimpur (Algorithmica 86 (2024), arXiv:2209.11209) 称一个集合族 $\mathcal{F}$ 为可塑的，如果满足：对于任意集合对 $A,B\in\mathcal{F}$，集合 $A\cap{B},A\cup{B},A-B,B-A$ 中至少有两个属于 $\mathcal{F}$。我们称一个可塑集合族 $\mathcal{F}$ 满足结构次模性，如果满足：对于任意交叉集合对 $A,B\in\mathcal{F}$，集合 $A\cap{B},A\cup{B}$ 中至少有一个属于 $\mathcal{F}$，并且集合 $A-B,B-A$ 中至少有一个属于 $\mathcal{F}$。对于任意正整数 $d\geq2$，我们构造了一个满足结构次模性的可塑集合族 $\mathcal{F}$，使得 (a) 不存在对称次模函数 $g$ 和 $λ\in{\mathbb Q}$ 满足 $\mathcal{F} = \{ S \,:\, g(S)<λ\}$，且 (b) $\mathcal{F}$ 不能被划分为 $d$ 个（或更少）不可交叉的族。