Hyperseparoids: a representation theorem (Q629848)

A \(k\)-separoid is a finite set \(S\) endowed with a symmetric \(k\)-relational system \(\mathcal{T}\subseteq 2^S\times\dots\times 2^S\; (k\)-times) defined in its family of subsets, such that for \(A_1, \dots, A_n\in S\) the following implications hold: (i) \(\{A_1, \dots, A_k\}\in \mathcal{T} \Rightarrow A_i\cap A_j=\emptyset\) whenever \(i\neq j\); (ii) \(\{A_1, \dots, A_k\}\in \mathcal{T}\) and \(B\subseteq (S\setminus \bigcup A_i) \Rightarrow \{A_1, \dots, A_k \cup B\}\in \mathcal{T}\). The elements of \(\mathcal{T}\) are called Tverberg partitions. The \(k\)-separoid is said to be acyclic if the components of each Tverberg partition are all nonempty. The main result is the following: Theorem 1. Evey acyclic \(3\)-separoid of order \(n=|S|\) can be represented by a family of convex polytopes, and their Tverberg partitions, in the \((n-1)\)-dimensional Euclidean space. At the end, in the spirit of Tverberg's theorem, a generalization of the concept of separoid, named hyperseparoid, is introduced. This means a set \(S\) endowed with a collection \(\mathcal{T}\) of finite subfamilies of \(S\) satisfying, for \(A_1, \dots, A_n\in S\) the previous conditions (i) and (ii) as well as the following condition: \(\{A_1, \dots, A_k\}\in \mathcal{T} \Rightarrow \{A_1, \dots, A_{k-1}\}\in \mathcal{T}\). A similar construction to that used in the proof of Theorem 1 leads to a representation of hyperseparoids by convex polytopes and their Tverberg partitions.