On the solvability of systems of linear equations over the ring $\mathbb{Z}$ of integers

Herrlich, Horst, and Tachtsis, Eleftherios. "On the solvability of systems of linear equations over the ring $\mathbb {Z}$ of integers." Commentationes Mathematicae Universitatis Carolinae 58.2 (2017): 241-260. <http://eudml.org/doc/288182>.

@article{Herrlich2017,
abstract = {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 $\mathsf \{ZFA\}$.},
author = {Herrlich, Horst, Tachtsis, Eleftherios},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Axiom of Choice; weak axioms of choice; linear equations with coefficients in $\mathbb \{Z\}$; infinite systems of linear equations over $\mathbb \{Z\}$; non-trivial solution of a system in $\mathbb \{Z\}$; permutation models of $\mathsf \{ZFA\}$; symmetric models of $\mathsf \{ZF\}$},
language = {eng},
number = {2},
pages = {241-260},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On the solvability of systems of linear equations over the ring $\mathbb \{Z\}$ of integers},
url = {http://eudml.org/doc/288182},
volume = {58},
year = {2017},
}

TY - JOUR
AU - Herrlich, Horst
AU - Tachtsis, Eleftherios
TI - On the solvability of systems of linear equations over the ring $\mathbb {Z}$ of integers
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2017
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 58
IS - 2
SP - 241
EP - 260
AB - 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 $\mathsf {ZFA}$.
LA - eng
KW - Axiom of Choice; weak axioms of choice; linear equations with coefficients in $\mathbb {Z}$; infinite systems of linear equations over $\mathbb {Z}$; non-trivial solution of a system in $\mathbb {Z}$; permutation models of $\mathsf {ZFA}$; symmetric models of $\mathsf {ZF}$
UR - http://eudml.org/doc/288182
ER -