Free matroid

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. 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.

Free extension of a matroid
The free extension of a matroid $$M$$ by some element $$e\not\in M$$, denoted $$M+e$$, is a matroid whose elements are the elements of $$M$$ plus the new element $$e$$, and:


 * Its circuits are the circuits of $$M$$ plus the sets $$B\cup \{e\}$$ for all bases $$B$$ of $$M$$.
 * Equivalently, its independent sets are the independent sets of $$M$$ plus the sets $$I\cup \{e\}$$ for all independent sets $$I$$ that are not bases.
 * Equivalently, its bases are the bases of $$M$$ plus the sets $$I\cup \{e\}$$ for all independent sets of size $$\text{rank}(M)-1$$.