Independence system

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In combinatorial mathematics, an independence system S is a pair (EI), where E is a finite set and I is a collection of subsets of E (called the independent sets) with the following properties:

  1. The empty set is independent, i.e., ∅ ∈ I. (Alternatively, at least one subset of E is independent, i.e., I ≠ ∅.)
  2. Every subset of an independent set is independent, i.e., for each  ⊆ X, X ∈ I →  ∈ I. This is sometimes called the hereditary property.

Adding the augmentation property or the independent set exchange property yields a matroid.

For a more general description, see abstract simplicial complex.