Semidirect sums of matroids (Q2343463)

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.