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:
- The empty set is independent, i.e., ∅ ∈ I. (Alternatively, at least one subset of V is independent, i.e., I ≠ ∅.)
- 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, or downward-closedness.
Another term for an independence system is an abstract simplicial complex.
Relation to other concepts
1. 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.
2. An independence system with an additional property called the augmentation property or the 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.