Independence system

In combinatorial mathematics, an independence system S is a pair (E, I), 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 Y  ⊆ X, we have X ∈ I → Y  ∈ 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.