In combinatorial mathematics, an independence system is a pair , where is a finite set and is a collection of subsets of (called the independent sets or feasible sets) with the following properties: The empty set is independent, i.e., . (Alternatively, at least one subset of is independent, i.e., .) Every subset of an independent set is independent, i.e., for each , we have . 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 A pair , where is a finite set and is a collection of subsets of is also called a hypergraph. When using this terminology, the elements in the set are called vertices and elements in the family 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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