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.
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N. A. Lebedinskaya, D. M. Lebedinskii, and A. A. Smirnov, “Possible dimensions of subspace intersections for five direct summands,” Zap. Nauchn. Semin. POMI, 453 189–197 (2016).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 463, 2017, pp. 269–276.
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Lebedinskaya, N.A., Lebedinskiǐ, D.M. & Smirnov, A.A. A Generalization of the Theorem on Forming a Matroid from Parts. J Math Sci 232, 921–925 (2018). https://doi.org/10.1007/s10958-018-3919-5
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DOI: https://doi.org/10.1007/s10958-018-3919-5