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.[1] 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 by some element , denoted , is a matroid whose elements are the elements of plus the new element , and:
Its
circuits are the circuits of plus the sets for all
bases of .[2]
Equivalently, its independent sets are the independent sets of plus the sets for all independent sets that are not bases.
Equivalently, its
bases are the bases of plus the sets for all independent sets of size .
References
^Oxley, James G. (2006). Matroid Theory. Oxford Graduate Texts in Mathematics. Vol. 3. Oxford University Press. p. 17.
ISBN9780199202508.