Independence system

In combinatorial mathematics, an independence system $S$ is a pair $(V,{ℐ})$, where $V$ is a finite set and ${ℐ}$ is a collection of subsets of $V$ (called the independent sets or feasible sets) with the following properties:

1. The empty set is independent, i.e., $\mathrm{\varnothing }\in {ℐ}$. (Alternatively, at least one subset of $V$ is independent, i.e., ${ℐ}\ne \mathrm{\varnothing }$.)
2. Every subset of an independent set is independent, i.e., for each $Y\subseteq X$, we have $X\in {ℐ}\Rightarrow Y\in {ℐ}$. This is sometimes called the hereditary property, or downward-closedness.

Another term for an independence system is an abstract simplicial complex.

Relation to other concepts

• A pair $(V,{ℐ})$, where $V$ is a finite set and ${ℐ}$ is a collection of subsets of $V$, is also called a hypergraph. When using this terminology, the elements in the set $V$ are called vertices and elements in the family ${ℐ}$ 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. The following expression summarizes the relations between the terms:

  HYPERGRAPHS ⊃ INDEPENDENCE-SYSTEMS = ABSTRACT-SIMPLICIAL-COMPLEXES ⊃ MATROIDS.

References

• Bondy, Adrian; Murty, U.S.R. (2008), Graph Theory, Graduate Texts in Mathematics, 244, Springer, p. 195, ISBN 9781846289699, https://books.google.com/books?id=HuDFMwZOwcsC&pg=PA195 .