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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2021
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2107.11700 |
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| _version_ | 1866913914386644992 |
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| author | Baker, Matthew Zhang, Tianyi |
| author_facet | Baker, Matthew Zhang, Tianyi |
| contents | Baker and Bowler defined a category of algebraic objects called tracts which generalize both partial fields and hyperfields. They also defined a notion of weak and strong matroids over a tract $F$, and proved that if $F$ is perfect, meaning that $F$-vectors and $F$-covectors are orthogonal for every matroid over $F$, then the notions of weak and strong $F$-matroids coincide. We define the class of strongly fused tracts and prove that such tracts are perfect. We in fact prove a more general result which implies that given a tract $F$, there is a tract $σ(F)$ with the same 3-term additive relations as $F$ such that weak $F$-matroids coincide with strong $σ(F)$-matroids. We also show that both partial fields and stringent hyperfields are strongly fused; in this way, our criterion for perfection generalizes results of Baker-Bowler and Bowler-Pendavingh. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2107_11700 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Fusion rules for pastures and tracts Baker, Matthew Zhang, Tianyi Combinatorics Baker and Bowler defined a category of algebraic objects called tracts which generalize both partial fields and hyperfields. They also defined a notion of weak and strong matroids over a tract $F$, and proved that if $F$ is perfect, meaning that $F$-vectors and $F$-covectors are orthogonal for every matroid over $F$, then the notions of weak and strong $F$-matroids coincide. We define the class of strongly fused tracts and prove that such tracts are perfect. We in fact prove a more general result which implies that given a tract $F$, there is a tract $σ(F)$ with the same 3-term additive relations as $F$ such that weak $F$-matroids coincide with strong $σ(F)$-matroids. We also show that both partial fields and stringent hyperfields are strongly fused; in this way, our criterion for perfection generalizes results of Baker-Bowler and Bowler-Pendavingh. |
| title | Fusion rules for pastures and tracts |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2107.11700 |