Semidirect sums of matroids
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2012
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| _version_ | 1866910556030500864 |
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| author | Bonin, Joseph E. Kung, Joseph P. S. |
| author_facet | Bonin, Joseph E. Kung, Joseph P. S. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1210_0626 |
| institution | arXiv |
| publishDate | 2012 |
| record_format | arxiv |
| spellingShingle | Semidirect sums of matroids Bonin, Joseph E. Kung, Joseph P. S. Combinatorics 05B35, 05B20, 05C35, 05D99, 06C10, 51M04 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. |
| title | Semidirect sums of matroids |
| topic | Combinatorics 05B35, 05B20, 05C35, 05D99, 06C10, 51M04 |
| url | https://arxiv.org/abs/1210.0626 |