Independence system

In combinatorial mathematics, an independence system $S$ is a pair $$(V, \mathcal{I})$$, where $V$ is a finite set and $\mathcal{I}$ is a collection of subsets of $V$ (called the independent sets or feasible sets) with the following properties: Another term for an independence system is an abstract simplicial complex.
 * 1) The empty set is independent, i.e., $$\emptyset\in\mathcal{I}$$.  (Alternatively, at least one subset of $V$ is independent, i.e., $$\mathcal{I}\neq\emptyset$$.)
 * 2) Every subset of an independent set is independent, i.e., for each $$Y\subseteq X$$, we have $$X\in\mathcal{I}\Rightarrow Y\in\mathcal{I}$$.  This is sometimes called the hereditary property, or downward-closedness.

Relation to other concepts

 * A pair $$(V, \mathcal{I})$$, where $V$ is a finite set and $\mathcal{I}$ 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 $\mathcal{I}$ 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.