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**1 - 10**of**10**### 2 subsets which is closed as a filter in the natural partial order induced by the inclusion

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### UNIVERSALITY OF SEPAROIDS

"... Abstract. A separoid is a symmetric relation † ⊂ `2 S´ defined on disjoint 2 pairs of subsets of a given set S such that it is closed as a filter in the canonical partial order induced by the inclusion (i.e., A † B � A ′ † B ′ ⇐ ⇒ A ⊆ A ′ and B ⊆ B ′). We introduce the notion of homomorphism as a ..."

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Abstract. A separoid is a symmetric relation † ⊂ `2 S´ defined on disjoint 2 pairs of subsets of a given set S such that it is closed as a filter in the canonical partial order induced by the inclusion (i.e., A † B � A ′ † B ′ ⇐ ⇒ A ⊆ A ′ and B ⊆ B ′). We introduce the notion of homomorphism as a map which preserve the so-called “minimal Radon partitions ” and show that separoids, endowed with these maps, admits an embedding from the category of all finite graphs. This proves that separoids constitute a countable universal partial order. Furthermore, by embedding also all hypergraphs (all set systems) into such a category, we prove a “stronger ” universality property. We further study some structural aspects of the category of separoids. We completely solve the density problem for (all) separoids as well as for separoids of points. We also generalise the classic Radon’s theorem in a categorical setting as well as Hedetniemi’s product conjecture (which can be proved for oriented matroids).

### Realisation of separoids and a Tverberg-type problem

"... A separoid is a symmetric relation † ⊂ ( 2 S) 2 defined on pairs of disjoint subsets which is closed as a filter in the natural partial order (i.e., A † B ≼ C † D ⇐ ⇒ A ⊆ C and B ⊆ D). We discus the Geometric Representation Theorem for separoids: every separoid (S, †) can be represented by a famil ..."

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A separoid is a symmetric relation † ⊂ ( 2 S) 2 defined on pairs of disjoint subsets which is closed as a filter in the natural partial order (i.e., A † B ≼ C † D ⇐ ⇒ A ⊆ C and B ⊆ D). We discus the Geometric Representation Theorem for separoids: every separoid (S, †) can be represented by a family of (convex) polytopes, and their Radon partitions, in the Euclidean space of dimension |S | − 1. Furthermore, we introduce a new kind of separoids’ morphisms — called chromomorphisms — which allow us to study Tverberg’s generalisation (1966) of Radon’s theorem (1921) in the context of convex sets. In particular the following Tverberg-type theorem is proved: Let S be a separoid of order |S | = (k − 1)(d(S) + 1) + 1, where d(S) denotes the (combinatorial) dimension of S. If there exists a monomorphism S → P into a separoid of points in general position in IE d, then there exists a chromomorphism S − → Kk onto the complete separoid of order k. This theorem is, in a sense, dual to the Hadwiger-type theorem proved by

### Two geometric representation theorems for separoids

"... Separoids capture the combinatorial structure which arises from the separations by hyper-planes of a family of convex sets in some Euclidian space. Furthermore, as we prove in this note, every abstract separoid S can be represented by a family of convex sets in the (|S | − 1)-dimensional Euclidian ..."

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Separoids capture the combinatorial structure which arises from the separations by hyper-planes of a family of convex sets in some Euclidian space. Furthermore, as we prove in this note, every abstract separoid S can be represented by a family of convex sets in the (|S | − 1)-dimensional Euclidian space. The geometric dimension of the separoid is the minimum dimen-sion where it can be represented and the upper bound given here is tight. Separoids have also the notions of combinatorial dimension and general position which are purely combinatorial in nature. In this note we also prove that: a separoid in general position can be represented by a family of points if and only if its geometric and combinatorial dimensions coincide. 1

### Tverberg-type theorems . . .

"... Let S be a d-dimensional separoid of (k − 1)(d + 1) + 1 convex sets in some ‘large-dimensional ’ Euclidean space IE N. We prove a theorem that can be interpreted as follows: if the separoid S can be mapped with a monomorphism to a d-dimensional separoid of points P in general position, then there ex ..."

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Let S be a d-dimensional separoid of (k − 1)(d + 1) + 1 convex sets in some ‘large-dimensional ’ Euclidean space IE N. We prove a theorem that can be interpreted as follows: if the separoid S can be mapped with a monomorphism to a d-dimensional separoid of points P in general position, then there exists a k-colouring ς: S → Kk such that, for each pair of colours i, j ∈ Kk, the convex hulls of their preimages do intersect —they are not separated. Here, by a monomorphism we mean an injective function such that the preimage of separated sets are separated. In a sense, this result is ‘dual’ to the Hadwiger-type theorems proved by Goodman & Pollack (1988) and Arocha, Bracho,