تقرير
Siblings of an $\aleph_0$-categorical relational structure
| العنوان: | Siblings of an $\aleph_0$-categorical relational structure |
|---|---|
| المؤلفون: | Laflamme, Claude, Pouzet, Maurice, Sauer, Norbert, Woodrow, Robert |
| سنة النشر: | 2018 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Logic, 06A, 06B |
| الوصف: | A sibling of a relational structure $R$ is any structure $S$ which can be embedded into $R$ and, vice versa, in which $R$ can be embedded. Let $sib(R)$ be the number of siblings of $R$, these siblings being counted up to isomorphism. Thomass\'e conjectured that for countable relational structures made of at most countably many relations, $sib(R)$ is either $1$, countably infinite, or the size of the continuum; but even showing the special case $sib(R)=1$ or infinite is unsettled when $R$ is a countable tree. This is related to Bonato-Tardif conjecture asserting that for every tree $T$ the number of trees which are sibling of $T$ is either one or infinite. We prove that if $R$ is countable and $\aleph_{0}$-categorical, then indeed $sib(R)$ is one or infinite. Furthermore, $sib(R)$ is one if and only if $R$ is finitely partitionable in the sense of Hodkinson and Macpherson. The key tools in our proof are the notion of monomorphic decomposition of a relational structure introduced in a paper by Pouzet and Thi\'ery 2013 and studied further by Oudrar and Pouzet 2015, and a result of Frasnay 1984. Comment: 34 pp |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/1811.04185 |
| رقم الأكسشن: | edsarx.1811.04185 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |