Wild generalised truncation of infinite matroids
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866912313405079552 |
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| author | Gollin, J. Pascal Joó, Attila |
| author_facet | Gollin, J. Pascal Joó, Attila |
| contents | For ${n \in \mathbb{N}}$, the $n$-truncation of a matroid $M$ of rank at least $n$ is the matroid whose bases are the $n$-element independent sets of $M$. One can extend this definition to negative integers by letting the $(-n)$-truncation be the matroid whose bases are all the sets that can be obtained by deleting $n$ elements of a base of $M$. If $M$ has infinite rank, then for distinct ${m,n \in \mathbb{Z}}$ the $m$-truncation and the $n$-truncation are distinct matroids.
Inspired by the work of Bowler and Geschke on infinite uniform matroids, we provide a natural definition of generalised truncations that encompasses the notions mentioned above. We call a generalised truncation wild if it is not an $n$-truncation for any ${n \in \mathbb{Z}}$ and we prove that, under Martin's Axiom, any finitary matroid of infinite rank and size of less than continuum admits ${2^{2^{\aleph_0}}}$ wild generalised truncations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_05064 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Wild generalised truncation of infinite matroids Gollin, J. Pascal Joó, Attila Combinatorics Logic 05B35 (primary) 03E50 (secondary) For ${n \in \mathbb{N}}$, the $n$-truncation of a matroid $M$ of rank at least $n$ is the matroid whose bases are the $n$-element independent sets of $M$. One can extend this definition to negative integers by letting the $(-n)$-truncation be the matroid whose bases are all the sets that can be obtained by deleting $n$ elements of a base of $M$. If $M$ has infinite rank, then for distinct ${m,n \in \mathbb{Z}}$ the $m$-truncation and the $n$-truncation are distinct matroids. Inspired by the work of Bowler and Geschke on infinite uniform matroids, we provide a natural definition of generalised truncations that encompasses the notions mentioned above. We call a generalised truncation wild if it is not an $n$-truncation for any ${n \in \mathbb{Z}}$ and we prove that, under Martin's Axiom, any finitary matroid of infinite rank and size of less than continuum admits ${2^{2^{\aleph_0}}}$ wild generalised truncations. |
| title | Wild generalised truncation of infinite matroids |
| topic | Combinatorics Logic 05B35 (primary) 03E50 (secondary) |
| url | https://arxiv.org/abs/2504.05064 |