In mathematics, a separoid is a binary relation between disjoint sets which is stable as an ideal in the canonical order induced by inclusion. Many mathematical objects which appear to be quite different, find a common generalisation in the framework of separoids; e.g., graphs, configurations of convex sets, oriented matroids, and polytopes. Any countable category is an induced subcategory of separoids when they are endowed with homomorphisms[1] (viz., mappings that preserve the so-called minimal Radon partitions).

In this general framework, some results and invariants of different categories turn out to be special cases of the same aspect; e.g., the pseudoachromatic number from graph theory and the Tverberg theorem from combinatorial convexity are simply two faces of the same aspect, namely, complete colouring of separoids.

The axioms

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A separoid[2] is a set   endowed with a binary relation   on its power set, which satisfies the following simple properties for  :

 
 
 

A related pair   is called a separation and we often say that A is separated from B. It is enough to know the maximal separations to reconstruct the separoid.

A mapping   is a morphism of separoids if the preimages of separations are separations; that is, for  

 

Examples

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Examples of separoids can be found in almost every branch of mathematics.[3][4][5] Here we list just a few.

1. Given a graph G=(V,E), we can define a separoid on its vertices by saying that two (disjoint) subsets of V, say A and B, are separated if there are no edges going from one to the other; i.e.,

 

2. Given an oriented matroid[5] M = (E,T), given in terms of its topes T, we can define a separoid on E by saying that two subsets are separated if they are contained in opposite signs of a tope. In other words, the topes of an oriented matroid are the maximal separations of a separoid. This example includes, of course, all directed graphs.

3. Given a family of objects in a Euclidean space, we can define a separoid in it by saying that two subsets are separated if there exists a hyperplane that separates them; i.e., leaving them in the two opposite sides of it.

4. Given a topological space, we can define a separoid saying that two subsets are separated if there exist two disjoint open sets which contains them (one for each of them).

The basic lemma

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Every separoid can be represented with a family of convex sets in some Euclidean space and their separations by hyperplanes.

References

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  1. ^ Strausz, Ricardo (1 March 2007). "Homomorphisms of separoids". Electronic Notes in Discrete Mathematics. 28: 461–468. doi:10.1016/j.endm.2007.01.064. Zbl 1291.05036.
  2. ^ Strausz, Ricardo (2005). "Separoids and a Tverberg-type problem". Geombinatorics. 15 (2): 79–92. Zbl 1090.52005.
  3. ^ Arocha, Jorge Luis; Bracho, Javier; Montejano, Luis; Oliveros, Deborah; Strausz, Ricardo (2002). "Separoids, their categories and a Hadwiger-type theorem for transversals". Discrete and Computational Geometry. 27 (3): 377–385. doi:10.1007/s00454-001-0075-2.
  4. ^ Nešetřil, Jaroslav; Strausz, Ricardo (2006). "Universality of separoids" (PDF). Archivum Mathematicum (Brno). 42 (1): 85–101.
  5. ^ a b Montellano-Ballesteros, Juan José; Strausz, Ricardo (July 2006). "A characterization of cocircuit graphs of uniform oriented matroids". Journal of Combinatorial Theory. Series B. 96 (4): 445–454. doi:10.1016/j.jctb.2005.09.008. Zbl 1109.52016.

Further reading

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