{"title":"Hindman 理想、Ramsey 理想和可求和理想之间的 Katětov 秩","authors":"Rafał Filipów, Krzysztof Kowitz, Adam Kwela","doi":"10.1007/s00153-024-00924-7","DOIUrl":null,"url":null,"abstract":"<div><p>A family <span>\\(\\mathcal {I}\\)</span> of subsets of a set <i>X</i> is an <i>ideal on X</i> if it is closed under taking subsets and finite unions of its elements. An ideal <span>\\(\\mathcal {I}\\)</span> on <i>X</i> is below an ideal <span>\\(\\mathcal {J}\\)</span> on <i>Y</i> in the <i>Katětov order</i> if there is a function <span>\\(f{: }Y\\rightarrow X\\)</span> such that <span>\\(f^{-1}[A]\\in \\mathcal {J}\\)</span> for every <span>\\(A\\in \\mathcal {I}\\)</span>. We show that the Hindman ideal, the Ramsey ideal and the summable ideal are pairwise incomparable in the Katětov order, where</p><ul>\n <li>\n <p>The <i>Ramsey ideal</i> consists of those sets of pairs of natural numbers which do not contain a set of all pairs of any infinite set (equivalently do not contain, in a sense, any infinite complete subgraph),</p>\n </li>\n <li>\n <p>The <i>Hindman ideal</i> consists of those sets of natural numbers which do not contain any infinite set together with all finite sums of its members (equivalently do not contain IP-sets that are considered in Ergodic Ramsey theory),</p>\n </li>\n <li>\n <p>The <i>summable ideal</i> consists of those sets of natural numbers such that the series of the reciprocals of its members is convergent.\n</p>\n </li>\n </ul></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"63 7-8","pages":"859 - 876"},"PeriodicalIF":0.3000,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-024-00924-7.pdf","citationCount":"0","resultStr":"{\"title\":\"Katětov order between Hindman, Ramsey and summable ideals\",\"authors\":\"Rafał Filipów, Krzysztof Kowitz, Adam Kwela\",\"doi\":\"10.1007/s00153-024-00924-7\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>A family <span>\\\\(\\\\mathcal {I}\\\\)</span> of subsets of a set <i>X</i> is an <i>ideal on X</i> if it is closed under taking subsets and finite unions of its elements. An ideal <span>\\\\(\\\\mathcal {I}\\\\)</span> on <i>X</i> is below an ideal <span>\\\\(\\\\mathcal {J}\\\\)</span> on <i>Y</i> in the <i>Katětov order</i> if there is a function <span>\\\\(f{: }Y\\\\rightarrow X\\\\)</span> such that <span>\\\\(f^{-1}[A]\\\\in \\\\mathcal {J}\\\\)</span> for every <span>\\\\(A\\\\in \\\\mathcal {I}\\\\)</span>. We show that the Hindman ideal, the Ramsey ideal and the summable ideal are pairwise incomparable in the Katětov order, where</p><ul>\\n <li>\\n <p>The <i>Ramsey ideal</i> consists of those sets of pairs of natural numbers which do not contain a set of all pairs of any infinite set (equivalently do not contain, in a sense, any infinite complete subgraph),</p>\\n </li>\\n <li>\\n <p>The <i>Hindman ideal</i> consists of those sets of natural numbers which do not contain any infinite set together with all finite sums of its members (equivalently do not contain IP-sets that are considered in Ergodic Ramsey theory),</p>\\n </li>\\n <li>\\n <p>The <i>summable ideal</i> consists of those sets of natural numbers such that the series of the reciprocals of its members is convergent.\\n</p>\\n </li>\\n </ul></div>\",\"PeriodicalId\":48853,\"journal\":{\"name\":\"Archive for Mathematical Logic\",\"volume\":\"63 7-8\",\"pages\":\"859 - 876\"},\"PeriodicalIF\":0.3000,\"publicationDate\":\"2024-05-03\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://link.springer.com/content/pdf/10.1007/s00153-024-00924-7.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Archive for Mathematical Logic\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s00153-024-00924-7\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"Arts and Humanities\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Archive for Mathematical Logic","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00153-024-00924-7","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"Arts and Humanities","Score":null,"Total":0}
引用次数: 0
摘要
如果一个集合 X 的子集的族（\mathcal {I}\）在取其元素的子集和有限联合下是封闭的，那么它就是 X 上的理想。如果存在一个函数 \(f{: }Y\rightarrow X\) 使得对于每一个 \(A\in \mathcal {I}\)来说，\(f^{-1}[A]\in \mathcal {J}\)都是 Y 上的理想 \(\mathcal {J}\)，那么 X 上的理想 \(\mathcal {I}\)在卡泰托夫阶中就是 Y 上的理想 \(\mathcal {J}\)的下面。我们证明，辛德曼理想、拉姆齐理想和可求和理想在卡泰托夫秩中是成对不可比的，其中拉姆齐理想由那些不包含任何无限集的所有对的集合的自然数对集合组成（在某种意义上等同于不包含任何无限完整子图）、可求和理想由自然数集组成，这些自然数集的成员的倒数序列是收敛的。
Katětov order between Hindman, Ramsey and summable ideals
A family \(\mathcal {I}\) of subsets of a set X is an ideal on X if it is closed under taking subsets and finite unions of its elements. An ideal \(\mathcal {I}\) on X is below an ideal \(\mathcal {J}\) on Y in the Katětov order if there is a function \(f{: }Y\rightarrow X\) such that \(f^{-1}[A]\in \mathcal {J}\) for every \(A\in \mathcal {I}\). We show that the Hindman ideal, the Ramsey ideal and the summable ideal are pairwise incomparable in the Katětov order, where
The Ramsey ideal consists of those sets of pairs of natural numbers which do not contain a set of all pairs of any infinite set (equivalently do not contain, in a sense, any infinite complete subgraph),
The Hindman ideal consists of those sets of natural numbers which do not contain any infinite set together with all finite sums of its members (equivalently do not contain IP-sets that are considered in Ergodic Ramsey theory),
The summable ideal consists of those sets of natural numbers such that the series of the reciprocals of its members is convergent.
期刊介绍：
The journal publishes research papers and occasionally surveys or expositions on mathematical logic. Contributions are also welcomed from other related areas, such as theoretical computer science or philosophy, as long as the methods of mathematical logic play a significant role. The journal therefore addresses logicians and mathematicians, computer scientists, and philosophers who are interested in the applications of mathematical logic in their own field, as well as its interactions with other areas of research.
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