| Title:
|
On the solvability of systems of linear equations over the ring $\mathbb{Z}$ of integers (English) |
| Author:
|
Herrlich, Horst |
| Author:
|
Tachtsis, Eleftherios |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
58 |
| Issue:
|
2 |
| Year:
|
2017 |
| Pages:
|
241-260 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We investigate the question whether a system $(E_i)_{i\in I}$ of homogeneous linear equations over $\mathbb{Z}$ is non-trivially solvable in $\mathbb{Z}$ provided that each subsystem $(E_j)_{j\in J}$ with $|J|\le c$ is non-trivially solvable in $\mathbb{Z}$ where $c$ is a fixed cardinal number such that $c< |I|$. Among other results, we establish the following. (a) The answer is `No' in the finite case (i.e., $I$ being finite). (b) The answer is `No' in the denumerable case (i.e., $|I|=\aleph_{0}$ and $c$ a natural number). (c) The answer in case that $I$ is uncountable and $c\le\aleph_{0}$ is `No relatively consistent with $\mathsf{ZF}$', but is unknown in $\mathsf{ZFC}$. For the above case, we show that ``every uncountable system of linear homogeneous equations over $\mathbb{Z}$, each of its countable subsystems having a non-trivial solution in $\mathbb{Z}$, has a non-trivial solution in $\mathbb{Z}$'' implies (1) the Axiom of Countable Choice (2) the Axiom of Choice for families of non-empty finite sets (3) the Kinna--Wagner selection principle for families of sets each order isomorphic to $\mathbb{Z}$ with the usual ordering, and is not implied by (4) the Boolean Prime Ideal Theorem ($\mathsf{BPI}$) in $\mathsf{ZF}$ (5) the Axiom of Multiple Choice ($\mathsf{MC}$) in $\mathsf{ZFA}$ (6) $\mathsf{DC}_{<\kappa}$ in $\mathsf{ZF}$, for every regular well-ordered cardinal number $\kappa$. We also show that the related statement ``every uncountable system of linear homogeneous equations over $\mathbb{Z}$, each of its countable subsystems having a non-trivial solution in $\mathbb{Z}$, has an uncountable subsystem with a non-trivial solution in $\mathbb{Z}$'' (1) is provable in $\mathsf{ZFC}$ (2) is not provable in $\mathsf{ZF}$ (3) does not imply ``every uncountable system of linear homogeneous equations over $\mathbb{Z}$, each of its countable subsystems having a non-trivial solution in $\mathbb{Z}$, has a non-trivial solution in $\mathbb{Z}$'' in $\mathsf{ZFA}$. (English) |
| Keyword:
|
Axiom of Choice |
| Keyword:
|
weak axioms of choice |
| Keyword:
|
linear equations with coefficients in $\mathbb{Z}$ |
| Keyword:
|
infinite systems of linear equations over $\mathbb{Z}$ |
| Keyword:
|
non-trivial solution of a system in $\mathbb{Z}$ |
| Keyword:
|
permutation models of $\mathsf{ZFA}$ |
| Keyword:
|
symmetric models of $\mathsf{ZF}$ |
| MSC:
|
03E25 |
| MSC:
|
03E35 |
| idZBL:
|
Zbl 06773717 |
| idMR:
|
MR3666944 |
| DOI:
|
10.14712/1213-7243.2015.207 |
| . |
| Date available:
|
2017-06-13T13:24:20Z |
| Last updated:
|
2020-01-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/146791 |
| . |
| Reference:
|
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| . |
