Independence system

In combinatorial mathematics, an independence system Template:Tmath is a pair $(V,{ℐ})$, where Template:Tmath is a finite set and Template:Tmath is a collection of subsets of Template:Tmath (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 Template:Tmath 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 Template:Tmath is a finite set and Template:Tmath is a collection of subsets of Template:Tmath, is also called a hypergraph. When using this terminology, the elements in the set Template:Tmath are called vertices and elements in the family Template:Tmath 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

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