---
tag: 8YPQ9EH87N
kind: concept
name: Free matroid
aliases: [free-extension]
sources:
  - wikipedia: en:65782984@r1283485257
    locator: Free matroid
    note: adopted
    mention: wikipedia://en/65782984
    sections: [Summary]
same_as:
  wikidata: Q104868806
  wikipedia: Free matroid
inventory: [HKYJJSTF4H]
---

## Summary

In mathematics, the **free matroid** over a given ground-set $E$ is the matroid in which the independent sets are all subsets of $E$. It is a special case of a uniform matroid; specifically, when $E$ has cardinality $n$, it is the uniform matroid $U{}^{n}_{n}$. The unique basis of this matroid is the ground-set itself, $E$. Among matroids on $E$, the free matroid on $E$ has the most independent sets, the highest rank, and the fewest circuits.