Extension equivalence of oriented matroids
By using a one-to-one correspondence between the point extensions of a matroid M and the set of all linear subclasses of M which, ordered by inclusion, form the so-called extension lattice of M, the authors introduce and investigate a special relation between matroids called extension equivalence. More precisely, they define two matroids to be extension equivalent if their extension lattices are isomorphic, and prove the following main results. First, they show that if M and M' are extension equivalent matroids having equal rank, then every adjoint of M is also an adjoint of M'. Second, using this result they prove that under the same assumption, M is representable over some field F if and only if M' is. And finally, the problem how to translate both the results to oriented matroids is discussed in the final section of the paper, and it is proved that they remain valid if the terms ``matroid'' and ``adjoint'' are replaced by the terms ``oriented matroid'' and ``oriented adjoint'', respectively.
scientific article; zbMATH DE number 4023307