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Independence system

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Source updated May 3, 2024

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:

The empty set is independent, i.e., $\emptyset \in {\mathcal {I}}$ . (Alternatively, at least one subset of ⁠ $V$ ⁠ is independent, i.e., ${\mathcal {I}}\neq \emptyset$ .)
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.

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

01Relation 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.

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This article is adapted from the Wikipedia article “Independence system”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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