Independence system

In combinatorial mathematics, an independence system ⁠$S$⁠ is a pair $(V,{\mathcal {I}})$, where ⁠$V$⁠ is a finite set and ⁠${\mathcal {I}}$⁠ 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., $\emptyset \in {\mathcal {I}}$. (Alternatively, at least one subset of ⁠$V$⁠ is independent, i.e., ${\mathcal {I}}\neq \emptyset$.)
2. Every subset of an independent set is independent, i.e., for each $Y\subseteq X$, we have $X\in {\mathcal {I}}\Rightarrow Y\in {\mathcal {I}}$. 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 $(V,{\mathcal {I}})$, where ⁠$V$⁠ is a finite set and ⁠${\mathcal {I}}$⁠ 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 ⁠${\mathcal {I}}$⁠ 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, vol. 244, Springer, p. 195, ISBN 9781846289699.