The free product of Matroids

Abstract:<BR> We introduce a noncommutative binary operation on matroids, called the free product, and discuss some of its properties. In particular, we show that the free product is characterized by a certain universal property, is associative, and respects duality. We characterize the minors of the free product, and prove that, up to isomorphism, every matroid factors uniquely into matroids that are irreducible with respect to free product. We use these results to prove an inequality involving the numbers of nonisomorphic matroids on n elements which was conjectured by Welsh in 1969, and to prove the freeness of the algebra of matroids whose product is dual to the restriction-contraction coproduct .