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.
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.
In other words, the topes of an oriented matroid are the maximal separations of a separoid.
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).
Every separoid can be represented with a family of convex sets in some Euclidean space and their separations by hyperplanes.