Independence system

In combinatorial mathematics, an independence system ⁠${\displaystyle S}$⁠ is a pair ${\displaystyle (V,{\mathcal {I}})}$, where ⁠${\displaystyle V}$⁠ is a finite set and ⁠${\displaystyle {\mathcal {I}}}$⁠ is a collection of subsets of ⁠${\displaystyle V}$⁠ (called the independent sets or feasible sets) with the following properties:

1. The empty set is independent, i.e., ${\displaystyle \emptyset \in {\mathcal {I}}}$. (Alternatively, at least one subset of ⁠${\displaystyle V}$⁠ is independent, i.e., ${\displaystyle {\mathcal {I}}\neq \emptyset }$.)
2. Every subset of an independent set is independent, i.e., for each ${\displaystyle Y\subseteq X}$, we have ${\displaystyle X\in {\mathcal {I}}\Rightarrow Y\in {\mathcal {I}}}$. This is sometimes called the hereditary property, or downward-closedness.

Another term for an independence system is an abstract simplicial complex.

• A pair ${\displaystyle (V,{\mathcal {I}})}$, where ⁠${\displaystyle V}$⁠ is a finite set and ⁠${\displaystyle {\mathcal {I}}}$⁠ is a collection of subsets of ⁠${\displaystyle V}$⁠, is also called a hypergraph. When using this terminology, the elements in the set ⁠${\displaystyle V}$⁠ are called vertices and elements in the family ⁠${\displaystyle {\mathcal {I}}}$⁠ 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.