Year: 2024
Type: article
Abstract: . We study the direct sum of [Formula: see text]-matroids by way of their cyclic flats. Using that the rank function of a [Formula: see text]-matroid is fully determined by the cyclic flats and their ranks, we show that the cyclic flats of the direct sum of two [Formula: see text]-matroids are exactly all the direct sums of the cyclic flats of the two summands. This simplifies the rank function of the direct sum significantly. A [Formula: see text]-matroid is called irreducible if it cannot be written as a (nontrivial) direct sum. We provide a characterization of irreducibility in terms of the cyclic flats and show that every [Formula: see text]-matroid can be decomposed into a direct sum of irreducible [Formula: see text]-matroids, which are unique up to equivalence.
DOI: 10.1137/23m156358x
Authors: Heide Gluesing-Luerssen, Benjamin Jany
Institutions: University of Kentucky, Eindhoven University of Technology
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Simons Foundation
Award: G2487157566
OpenAlex ID: W4404964210