The free product of matroids

We introduce a noncommutative binary operation on matroids, called free product. We show that this operation respects matroid duality, and has the property that, given only the cardinalities, an ordered pair of matroids may be recovered, up to isomorphism, from its free product. We use these results to give a short proof of Welsh’s 1969 conjecture, which provides a progressive lower bound for the number of isomorphism classes of matroids on an n-element set. In the minor coalgebra of matroids ([3], [1]), the coproduct of a matroid M(S) is given by ∑ A⊆S M |A⊗M/A, where M |A denotes the restriction of M to A and M/A denotes the matroid on the set difference S\A obtained by contracting A from M . The product of matroids M and N in the dual algebra is thus a linear combination ∑ L αLL of those matroids L having some restriction isomorphic to M , with complementary contraction isomorphic to N . The coefficient αL of L = L(U) is the number of subsets A ⊆ U such that L|A ∼= M and L/A ∼= N . If the matroids having nonzero coefficient in the product of M and N are ordered in the weak-map order, there is a final term equal to a scalar multiple of the direct product M ⊕N , and an initial term equal to a scalar multiple of a matroid that we have elected to call the free product of M and N . In the present short article we give an intrinsic definition of the free product of matroids, and prove the crucial result that, given only their cardinalities, the two factors themselves, and even the order of the factors, can be recovered, up to isomorphism, from the free product. This is in sharp contrast to the behavior of direct sums, where the failure of unique ordered factorization gave rise to a little crisis in matroid theory, holding up the proof of Welsh’s “self-evident” conjecture [4] for more than three decades. He conjectured that if there are fn isomorphism classes of matroids on an n-element set, then fn · fm ≤ fn+m, for all n,m ≥ 0. Where direct sum fails, free product succeeds; we prove the conjecture here. In future work we shall investigate in detail the combinatorial properties of the free product, as well as its implications for the minor coalgebra of matroids. We denote the rank and nullity functions of a matroid M(S) by ρM and νM , respectively, and denote by λM the rank-lack function on M , given by λM(A) = ρ(M)− ρM(A), for all A ⊆ S, where ρ(M) = ρM(S) is the rank of M . We denote the disjoint union of sets S and T by S + T and the intersection S ∩ T by either ST or TS. We refer the reader to Oxley’s book [2] for any background on matroid theory that might be needed. 2000 Mathematics Subject Classification. 05B35, 16W30, 05A15.