Abstract
For matroids M and N on disjoint sets S and T, a semidirect sum of M and N is any matroid K on \({S \cup T}\) that, like the direct sum and the free product, has the restriction \({K|S}\) equal to M and the contraction K/S equal to N. We abstract a matrix construction to get a general matroid construction: the matroid union of any rank-preserving extension of M on the set \({S \cup T}\) with the direct sum of N and the rank-0 matroid on S is a semidirect sum of M and N. We study principal sums in depth; these are such matroid unions where the extension of M has each element of T added either as a loop or freely on a fixed flat of M. A second construction of semidirect sums, defined by a Higgs lift, also specializes to principal sums. We also explore what can be deduced if M and N, or certain of their semidirect sums, are transversal or fundamental transversal matroids.
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To James Oxley on his 60th birthday
Supported by the National Security Agency under grant H98230-11-1-0183.
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Bonin, J.E., Kung, J.P.S. Semidirect Sums of Matroids. Ann. Comb. 19, 7–27 (2015). https://doi.org/10.1007/s00026-015-0253-1
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DOI: https://doi.org/10.1007/s00026-015-0253-1