Initial Tukey structure below a stable ordered-union ultrafilter
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912060623814656 |
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| author | Özalp, Tan |
| author_facet | Özalp, Tan |
| contents | Answering a question of Dobrinen and Todorcevic, we prove that below any stable ordered-union ultrafilter $\mathcal{U}$, there are exactly four nonprincipal Tukey classes: $[\mathcal{U}], [\mathcal{U}_{\operatorname{min}}], [\mathcal{U}_{\operatorname{max}}]$, and $[\mathcal{U}_{\operatorname{minmax}}]$. This parallels the classification of ultrafilters Rudin-Keisler below $\mathcal{U}$ by Blass. A key step in the proof involves modifying the proof of a canonization theorem of Klein and Spinas for Borel functions on $\mathrm{FIN}^{[\infty]}$ to obtain a simplified canonization theorem for fronts on $\mathrm{FIN}^{[\infty]}$, recovering Lefmann's canonization for fronts of finite uniformity rank as a special case. We use this to classify the Rudin-Keisler classes of all ultrafilters Tukey below $\mathcal{U}$, which is then applied to achieve the main result. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_04326 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Initial Tukey structure below a stable ordered-union ultrafilter Özalp, Tan Logic 03E05, 03E04, 05D10, 06A07 (Primary) 03E02, 54H05, 54D80 (Secondary) Answering a question of Dobrinen and Todorcevic, we prove that below any stable ordered-union ultrafilter $\mathcal{U}$, there are exactly four nonprincipal Tukey classes: $[\mathcal{U}], [\mathcal{U}_{\operatorname{min}}], [\mathcal{U}_{\operatorname{max}}]$, and $[\mathcal{U}_{\operatorname{minmax}}]$. This parallels the classification of ultrafilters Rudin-Keisler below $\mathcal{U}$ by Blass. A key step in the proof involves modifying the proof of a canonization theorem of Klein and Spinas for Borel functions on $\mathrm{FIN}^{[\infty]}$ to obtain a simplified canonization theorem for fronts on $\mathrm{FIN}^{[\infty]}$, recovering Lefmann's canonization for fronts of finite uniformity rank as a special case. We use this to classify the Rudin-Keisler classes of all ultrafilters Tukey below $\mathcal{U}$, which is then applied to achieve the main result. |
| title | Initial Tukey structure below a stable ordered-union ultrafilter |
| topic | Logic 03E05, 03E04, 05D10, 06A07 (Primary) 03E02, 54H05, 54D80 (Secondary) |
| url | https://arxiv.org/abs/2410.04326 |