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.^[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 edit

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 $M$ .^[2]
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 ${\text{rank}}(M)-1$ .

References edit

^ Oxley, James G. (2006). Matroid Theory. Oxford Graduate Texts in Mathematics. Vol. 3. Oxford University Press. p. 17. ISBN 9780199202508.
^ Bonin, Joseph E.; de Mier, Anna (2008). "The lattice of cyclic flats of a matroid". Annals of Combinatorics. 12 (2): 155–170. arXiv:math/0505689. doi:10.1007/s00026-008-0344-3.

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[1] Oxley, James G. (2006). Matroid Theory. Oxford Graduate Texts in Mathematics. Vol. 3. Oxford University Press. p. 17. ISBN 9780199202508.

[2] Bonin, Joseph E.; de Mier, Anna (2008). "The lattice of cyclic flats of a matroid". Annals of Combinatorics. 12 (2): 155–170. arXiv:math/0505689. doi:10.1007/s00026-008-0344-3.

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