Semidirect sums of matroids

TL;DR

This paper introduces a general construction for semidirect sums of matroids, explores principal sums in detail, and investigates properties related to transversal and fundamental transversal matroids.

Contribution

It provides a matrix-inspired construction for semidirect sums, studies principal sums extensively, and connects these concepts to transversal matroid properties.

Findings

A general matroid construction for semidirect sums via matroid union.

Principal sums are characterized by adding elements as loops or freely on flats.

Connections established between semidirect sums and transversal matroid properties.

Abstract

For matroids M and N on disjoint sets S and T, a semidirect sum of M and N is a matroid K on the union of S and T that, like the direct sum and the free product, has the restriction of K to S equal to M and the contraction of K to T 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 union of S and 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.