mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a separoid is a

binary relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of ele ...

between

disjoint sets In mathematics, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set.. For example, and are ''disjoint sets,'' while and are not disjoint. A ...

which is stable as an

ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...

in the canonical order induced by

inclusion Inclusion or Include may refer to: Sociology * Social inclusion, aims to create an environment that supports equal opportunity for individuals and groups that form a society. ** Inclusion (disability rights), promotion of people with disabiliti ...

. Many mathematical objects which appear to be quite different, find a common generalisation in the framework of separoids; e.g.,

graphs Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...

, configurations of

convex set In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex r ...

oriented matroid An oriented matroid is a mathematical structure that abstracts the properties of directed graphs, vector arrangements over ordered fields, and hyperplane arrangements over ordered fields. In comparison, an ordinary (i.e., non-oriented) matroid a ...

s, and

polytopes In elementary geometry, a polytope is a geometric object with flat sides (''faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an - ...

. Any countable

category Category, plural categories, may refer to: Philosophy and general uses * Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) * ...

is an induced subcategory of separoids when they are endowed with

homomorphism In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...

s (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

A separoid is a

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

S

endowed with a binary relation

\mid\ \subseteq2^S\times2^S

on its

power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...

, which satisfies the following simple properties for

A,B\subseteq S

: :

A\mid B\Leftrightarrow B\mid A,

A\mid B\Rightarrow A\cap B=\varnothing,

A\mid B \hbox A'\subset A\Rightarrow A'\mid B.

A related pair

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

\varphi\colon S\to T

is a

morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...

of separoids if the preimages of separations are separations; that is, for

A,B\subseteq T

A\mid B\Rightarrow\varphi^(A)\mid\varphi^(B).

Examples

Examples of separoids can be found in almost every branch of

. Here we list just a few. 1. Given a

graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...

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., :

A\mid B\Leftrightarrow\forall a\in A\hboxb\in B\colon ab\not\in E.

2. Given an oriented matroid ''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 graph In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs. Definition In formal terms, a directed graph is an ordered pa ...

s. 3. Given a family of objects in a

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...

, we can define a separoid in it by saying that two subsets are separated if there exists a

hyperplane In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...

that ''separates'' them; i.e., leaving them in the two opposite sides of it. 4. Given a

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...

, we can define a separoid saying that two subsets are separated if there exist two disjoint

open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suf ...

s 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.

The axioms

Examples

The basic lemma

References

Further reading