Five different ways of combinatorial description of non-empty faces of the cone of supermodular functions on the power set of a finite basic set $N$ are introduced. Their identification with faces of the cone of supermodular games allows one to associate to them certain polytopes in $\mathbb{R}^{N}$, known as cores (of these games) in context of cooperative game theory, or generalized permutohedra in context of polyhedral geometry. Non-empty faces of the supermodular cone then correspond to normal fans of those polytopes. This (basically) geometric way of description of faces of the cone then leads to the combinatorial ways of their description. The first combinatorial way is to identify the faces with certain partitions of the set of enumerations of $N$, known as rank tests in context of algebraic statistics. The second combinatorial way is to identify faces with certain collections of posets on $N$, known as (complete) fans of posets in context of polyhedral geometry. The third combinatorial way is to identify the faces with certain coverings of the power set of $N$, introduced relatively recently in context of cooperative game theory under name core structures. The fourth combinatorial way is to identify the faces with certain formal conditional independence structures, introduced formerly in context of multivariate statistics under name structural semi-graphoids. The fifth way is to identify the faces with certain subgraphs of the permutohedral graph, whose nodes are enumerations of $N$. We prove the equivalence of those six ways of description of non-empty faces of the supermodular cone. This result also allows one to describe the faces of the polyhedral cone of (rank functions of) polymatroids over $N$ and the faces of the submodular cone over $N$.

翻译：本文提出了五种组合描述方法，用于刻画定义在有限基集$N$幂集上的超模函数锥的非空面。通过将其与超模博弈锥的面进行对应，可将这些面关联到$\mathbb{R}^{N}$中的特定多面体——在合作博弈论背景下称为（博弈的）核心，在多面体几何学背景下则称为广义排列多面体。超模锥的非空面对应于这些多面体的正规扇。这种（本质上的）几何描述方式进而引出了面的组合描述方法。第一种组合方法是将面与$N$枚举集合的特定划分相对应，该划分在代数统计学中称为秩检验。第二种方法是将面与$N$上偏序集的特定集合相对应，该集合在多面体几何学中称为（完备）偏序集扇。第三种方法是将面与$N$幂集的特定覆盖相对应，该覆盖在合作博弈论中近期被提出，称为核心结构。第四种方法是将面与特定的形式条件独立结构相对应，该结构在多变量统计学中早先被提出，称为结构半图胚。第五种方法是将面与排列图（其节点为$N$的枚举）的特定子图相对应。我们证明了上述六种超模锥非空面描述方法的等价性。该结论还可用于描述$N$上多拟阵（秩函数）多面体锥的面以及$N$上子模锥的面。