Maximum linearizations of lower sets in $\mathbb{N}^m$ with application to monomial ideals
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2019
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| _version_ | 1866916725133410304 |
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| author | Altman, Harry Weiermann, Andreas |
| author_facet | Altman, Harry Weiermann, Andreas |
| contents | We compute the type (maximum linearization) of the well partial order of bounded lower sets in $\mathbb{N}^m$, ordered under inclusion, and find it is $ω^{ω^{m-1}}$. Moreover we compute the type of the set of all lower sets in $\mathbb{N}^m$, a topic studied by Aschenbrenner and Pong, and find that it is equal to \[ ω^{\sum_{k=1}^{m} ω^{m-k}\binom{m}{k-1} }+ 1. \] As a consequence we deduce corresponding bounds on effectively given sequences of monomial ideals in $F[X,Y]$ where $F$ is a field. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1909_06719 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Maximum linearizations of lower sets in $\mathbb{N}^m$ with application to monomial ideals Altman, Harry Weiermann, Andreas Logic 03E10 We compute the type (maximum linearization) of the well partial order of bounded lower sets in $\mathbb{N}^m$, ordered under inclusion, and find it is $ω^{ω^{m-1}}$. Moreover we compute the type of the set of all lower sets in $\mathbb{N}^m$, a topic studied by Aschenbrenner and Pong, and find that it is equal to \[ ω^{\sum_{k=1}^{m} ω^{m-k}\binom{m}{k-1} }+ 1. \] As a consequence we deduce corresponding bounds on effectively given sequences of monomial ideals in $F[X,Y]$ where $F$ is a field. |
| title | Maximum linearizations of lower sets in $\mathbb{N}^m$ with application to monomial ideals |
| topic | Logic 03E10 |
| url | https://arxiv.org/abs/1909.06719 |