The Nikodym and Grothendieck properties of Boolean algebras and rings related to ideals
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918475024302080 |
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| author | Sobota, Damian Żuchowski, Tomasz |
| author_facet | Sobota, Damian Żuchowski, Tomasz |
| contents | For an ideal $\mathcal{I}$ in a $σ$-complete Boolean algebra $\mathcal{A}$, we show that if the Boolean algebra $\mathcal{A}\langle\mathcal{I}\rangle$ generated by $\mathcal{I}$ does not have the Nikodym property, then it does not have the Grothendieck property either. The converse however does not hold -- we construct a family of $\mathfrak{c}$ many pairwise non-isomorphic Boolean subalgebras of the power set $\wp(ω)$ of the form $\wp(ω)\langle\mathcal{I}\rangle$ which, when thought of as subsets of the Cantor space $2^ω$, belong to the Borel class $\mathbb{F}_{σδ}$ and have the Nikodym property but not the Grothendieck property, and a family of $2^\mathfrak{c}$ many pairwise non-isomorphic non-analytic Boolean algebras of the form $\wp(ω)\langle\mathcal{I}\rangle$ with the Nikodym property but without the Grothendieck property.
Extending a result of Hernández-Hernández and Hrušák, we show that for an analytic P-ideal $\mathcal{I}$ on $ω$ the following are equivalent: 1) $\mathcal{I}$ is totally bounded, 2) $\mathcal{I}$ has the Local-to-Global Boundedness Property for submeasures, 3) $\wp(ω)/\mathcal{I}$ contains a countable splitting family, 4) $\mbox{conv}\le_K\mathcal{I}$. Moreover, proving a conjecture of Drewnowski, Florencio, and Paúl, we present examples of analytic P-ideals on $ω$ with the Nikodym property but without the Local-to-Global Boundedness Property for submeasures (and so not totally bounded). Exploiting a construction of Alon, Drewnowski, and Łuczak, we also describe a family of $\mathfrak{c}$ many pairwise non-isomorphic ideals on $ω$, induced by sequences of Kneser hypergraphs, which all have the Nikodym property but not the Nested Partition Property -- this answers a question of Stuart.
Finally, Tukey reducibility of a class of ideals without the Nikodym property is studied. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_19744 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Nikodym and Grothendieck properties of Boolean algebras and rings related to ideals Sobota, Damian Żuchowski, Tomasz Logic Functional Analysis General Topology For an ideal $\mathcal{I}$ in a $σ$-complete Boolean algebra $\mathcal{A}$, we show that if the Boolean algebra $\mathcal{A}\langle\mathcal{I}\rangle$ generated by $\mathcal{I}$ does not have the Nikodym property, then it does not have the Grothendieck property either. The converse however does not hold -- we construct a family of $\mathfrak{c}$ many pairwise non-isomorphic Boolean subalgebras of the power set $\wp(ω)$ of the form $\wp(ω)\langle\mathcal{I}\rangle$ which, when thought of as subsets of the Cantor space $2^ω$, belong to the Borel class $\mathbb{F}_{σδ}$ and have the Nikodym property but not the Grothendieck property, and a family of $2^\mathfrak{c}$ many pairwise non-isomorphic non-analytic Boolean algebras of the form $\wp(ω)\langle\mathcal{I}\rangle$ with the Nikodym property but without the Grothendieck property. Extending a result of Hernández-Hernández and Hrušák, we show that for an analytic P-ideal $\mathcal{I}$ on $ω$ the following are equivalent: 1) $\mathcal{I}$ is totally bounded, 2) $\mathcal{I}$ has the Local-to-Global Boundedness Property for submeasures, 3) $\wp(ω)/\mathcal{I}$ contains a countable splitting family, 4) $\mbox{conv}\le_K\mathcal{I}$. Moreover, proving a conjecture of Drewnowski, Florencio, and Paúl, we present examples of analytic P-ideals on $ω$ with the Nikodym property but without the Local-to-Global Boundedness Property for submeasures (and so not totally bounded). Exploiting a construction of Alon, Drewnowski, and Łuczak, we also describe a family of $\mathfrak{c}$ many pairwise non-isomorphic ideals on $ω$, induced by sequences of Kneser hypergraphs, which all have the Nikodym property but not the Nested Partition Property -- this answers a question of Stuart. Finally, Tukey reducibility of a class of ideals without the Nikodym property is studied. |
| title | The Nikodym and Grothendieck properties of Boolean algebras and rings related to ideals |
| topic | Logic Functional Analysis General Topology |
| url | https://arxiv.org/abs/2510.19744 |