| Title:
|
On the nontrivial solvability of systems of homogeneous linear equations over $\mathbb Z$ in ZFC (English) |
| Author:
|
Šaroch, Jan |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
61 |
| Issue:
|
2 |
| Year:
|
2020 |
| Pages:
|
155-164 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Motivated by the paper by H. Herrlich, E. Tachtsis (2017) we investigate in ZFC the following compactness question: for which uncountable cardinals $\kappa$, an arbitrary nonempty system $S$ of homogeneous $\mathbb Z$-linear equations is nontrivially solvable in $\mathbb Z$ provided that each of its subsystems of cardinality less than $\kappa$ is nontrivially solvable in $\mathbb Z$? (English) |
| Keyword:
|
homogeneous $\mathbb Z$-linear equation |
| Keyword:
|
$\kappa$-free group |
| Keyword:
|
$\mathcal L_{\omega_1\omega}$-compact cardinal |
| MSC:
|
03E35 |
| MSC:
|
03E55 |
| MSC:
|
08A45 |
| MSC:
|
13C10 |
| MSC:
|
20K30 |
| idZBL:
|
Zbl 07285998 |
| idMR:
|
MR4143702 |
| DOI:
|
10.14712/1213-7243.2020.017 |
| . |
| Date available:
|
2020-10-13T13:08:17Z |
| Last updated:
|
2022-07-04 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148280 |
| . |
| Reference:
|
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| Reference:
|
[2] Bagaria J., Magidor M.: On $\omega_1$-strongly compact cardinals.J. Symb. Log. 79 (2014), no. 1, 266–278. MR 3226024, 10.1017/jsl.2013.12 |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
[6] Göbel R., Trlifaj J.: Approximations and Endomorphism Algebras of Modules, Volume 1., Approximations.De Gruyter Expositions in Mathematics, 41, Walter de Gruyter GmbH & Co. KG, Berlin, 2012. MR 2985554 |
| Reference:
|
[7] Herrlich H., Tachtsis E.: On the solvability of systems of linear equations over the ring $\mathbb Z$ of integers.Comment. Math. Univ. Carolin. 58 (2017), no. 2, 241–260. MR 3666944 |
| Reference:
|
[8] Kanamori A.: The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings.Springer Monographs in Mathematics, Springer, Berlin, 2003. MR 1994835 |
| Reference:
|
[9] Shelah S.: Quite free complicated abelian group, PCF and black boxes.available at ArXiv: 1404.2775v2 [math.LO] (2019), 49 pages. |
| . |
