combinatorial Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many app ...

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 ...

, an independence system is a pair

(V, \mathcal)

, where is a finite

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 ...

and is a collection of

subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...

s of (called the independent sets or feasible sets) with the following properties: # The

empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...

is independent, i.e.,

\emptyset\in\mathcal

. (Alternatively, at least one subset of is independent, i.e.,

\mathcal\neq\emptyset

.) # Every subset of an independent set is independent, i.e., for each

Y\subseteq X

, we have

X\in\mathcal\Rightarrow Y\in\mathcal

. This is sometimes called the hereditary property, or downward-closedness. Another term for an independence system is an

abstract simplicial complex In combinatorics, an abstract simplicial complex (ASC), often called an abstract complex or just a complex, is a family of sets that is closed under taking subsets, i.e., every subset of a set in the family is also in the family. It is a purely c ...

Relation to other concepts

* A pair

(V, \mathcal)

, where is a finite

and is a collection of

s of is also called a

hypergraph In mathematics, a hypergraph is a generalization of a graph in which an edge can join any number of vertices. In contrast, in an ordinary graph, an edge connects exactly two vertices. Formally, an undirected hypergraph H is a pair H = (X,E) wh ...

. When using this terminology, the elements in the set are called vertices and elements in the family are called hyperedges. So an independence system can be defined shortly as a downward-closed hypergraph. * An independence system with an additional property called the augmentation property or the independent set exchange property yields a

matroid In combinatorics, a branch of mathematics, a matroid is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid axiomatically, the most significant being ...

. The following expression summarizes the relations between the terms:

HYPERGRAPHS INDEPENDENCE-SYSTEMS ABSTRACT-SIMPLICIAL-COMPLEXES MATROIDS.

References

*. Combinatorics Hypergraphs {{combin-stub