Separoid

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In

A separoid[2] is a set ${\displaystyle S}$ endowed with a binary relation ${\displaystyle \mid \ \subseteq 2^{S}\times 2^{S}}$ on its power set, which satisfies the following simple properties for ${\displaystyle A,B\subseteq S}$:

${\displaystyle A\mid B\Leftrightarrow B\mid A,}$

${\displaystyle A\mid B\Rightarrow A\cap B=\varnothing ,}$

${\displaystyle A\mid B{\hbox{ and }}A'\subset A\Rightarrow A'\mid B.}$

A related pair ${\displaystyle A\mid B}$ 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 ${\displaystyle \varphi \colon S\to T}$ is a morphism of separoids if the preimages of separations are separations; that is, for ${\displaystyle A,B\subseteq T}$

${\displaystyle A\mid B\Rightarrow \varphi ^{-1}(A)\mid \varphi ^{-1}(B).}$

Examples

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

1. Given a

${\displaystyle A\mid B\Leftrightarrow \forall a\in A{\hbox{ and }}b\in B\colon ab\not \in E.}$

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

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

References