Geodesic
A generalization of the theorem on forming a matroid from parts
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXX, Tome 463 (2017), pp. 269-276 
N. A. Lebedinskaya; D. M. Lebedinskii; A. A. Smirnov. A generalization of the theorem on forming a matroid from parts. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXX, Tome 463 (2017), pp. 269-276. http://geodesic.mathdoc.fr/item/ZNSL_2017_463_a16/
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

A generalization of the theorem on forming a matroid from parts is proved, i.e., given a finite set subdivided into some blocks, each of which is supplied with a matroid structure, and assuming that the ranks of every union of certain blocks are prescribed in such a way that the conditions on the rank function of a matroid are fulfilled, one can extend the rank function to all the subsets of the original set in such a way that the latter becomes a matroid.

[1] N. A. Lebedinskaya, D. M. Lebedinskii, A. A. Smirnov, “O vozmozhnykh znacheniyakh razmernostei peresechenii podprostranstv dlya pyati pryamykh slagaemykh”, Zap. nauchn. semin. POMI, 453, 2016, 189–197 | MR