The free product of $q$-matroids
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909431662379008 |
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| author | Alfarano, Gianira N. Byrne, Eimear Fulcher, Andrew |
| author_facet | Alfarano, Gianira N. Byrne, Eimear Fulcher, Andrew |
| contents | We introduce the notion of the free product of $q$-matroids, which is the $q$-analogue of the free product of matroids. We study the properties of this noncommutative binary operation, making an extensive use of the theory of cyclic flats. We show that the free product of two $q$-matroids $M_1$ and $M_2$ is maximal with respect to the weak order on $q$-matroids having $M_1$ as a restriction and $M_2$ as the complementary contraction. We characterise $q$-matroids that are irreducible with respect to the free product and we prove that the factorization of a $q$-matroid into a free product of irreducibles is unique up to isomorphism. We discuss the representability of the free product, with a particular focus on rank one uniform $q$-matroids and show that such a product is represented by clubs on the projective line. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_13025 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The free product of $q$-matroids Alfarano, Gianira N. Byrne, Eimear Fulcher, Andrew Combinatorics Information Theory 06A07, 06C20 We introduce the notion of the free product of $q$-matroids, which is the $q$-analogue of the free product of matroids. We study the properties of this noncommutative binary operation, making an extensive use of the theory of cyclic flats. We show that the free product of two $q$-matroids $M_1$ and $M_2$ is maximal with respect to the weak order on $q$-matroids having $M_1$ as a restriction and $M_2$ as the complementary contraction. We characterise $q$-matroids that are irreducible with respect to the free product and we prove that the factorization of a $q$-matroid into a free product of irreducibles is unique up to isomorphism. We discuss the representability of the free product, with a particular focus on rank one uniform $q$-matroids and show that such a product is represented by clubs on the projective line. |
| title | The free product of $q$-matroids |
| topic | Combinatorics Information Theory 06A07, 06C20 |
| url | https://arxiv.org/abs/2412.13025 |