Mathematics – Combinatorics
Scientific paper
2004-09-07
Mathematics
Combinatorics
Dedicated to Denis Higgs. 25 pages, 3 figures. Submitted for publication in the Journal of Combinatorial Theory (A). See arX
Scientific paper
We study the combinatorial, algebraic and geometric properties of the free product operation on matroids. After giving cryptomorphic definitions of free product in terms of independent sets, bases, circuits, closure, flats and rank function, we show that free product, which is a noncommutative operation, is associative and respects matroid duality. The free product of matroids $M$ and $N$ is maximal with respect to the weak order among matroids having $M$ as a submatroid, with complementary contraction equal to $N$. Any minor of the free product of $M$ and $N$ is a free product of a repeated truncation of the corresponding minor of $M$ with a repeated Higgs lift of the corresponding minor of $N$. We characterize, in terms of their cyclic flats, matroids that are irreducible with respect to free product, and prove that the factorization of a matroid into a free product of irreducibles is unique up to isomorphism. We use these results to determine, for K a field of characteristic zero, the structure of the minor coalgebra $\cal C$ of a family of matroids $\cal M$ that is closed under formation of minors and free products: namely, $\cal C$ is cofree, cogenerated by the set of irreducible matroids belonging to $\cal M$.
Crapo Henry
Schmitt William
No associations
LandOfFree
A unique factorization theorem for matroids does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with A unique factorization theorem for matroids, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and A unique factorization theorem for matroids will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-533188