A Generalization of the Theorem on Forming a Matroid from Parts

The following generalization of the theorem on forming a matroid from parts is proved: If a finite set is subdivided into some blocks, each of which is supplied with a matroid structure, and the ranks of every union of certain blocks are prescribed and satisfy the conditions on the rank function of a matroid, then the rank function can be extended to all the subsets of the original set in such a way that the latter becomes a matroid. © 2018, Springer Science+Business Media, LLC, part of Springer Nature.