On the nontrivial solvability of systems of homogeneous linear equations over $\mathbb Z$ in ZFC

TL;DR

This paper explores the conditions under which uncountable systems of homogeneous linear equations over integers are nontrivially solvable in ZFC, focusing on the compactness property relative to cardinalities.

Contribution

It investigates the nontrivial solvability of large systems of linear equations over integers within ZFC, extending previous work by Herrlich and Tachtsis.

Findings

Identifies specific uncountable cardinals where compactness holds

Establishes conditions for nontrivial solutions in large systems

Provides new insights into the structure of solutions over $\

Abstract

Following a recent paper by Herrlich and Tachtsis, we investigate in ZFC the following compactness question: for which unountable cardinals $\kappa$, an arbitrary nonempty system $S$ of homogeneous $\mathbb Z$-linear equations is nontrivially solvable in $\mathbb Z$ provided that each its nonempty subsystem of cardinality $<\kappa$ is nontrivially solvable in $\mathbb Z$?

Equations

$$b_{n\alpha }\psi (g_n)\ne z_{\alpha }-\sum _{k\in I_{\alpha }\setminus \{n\}}{b}_{k\alpha }\psi (g_k).$$

$$M_{\alpha }\cap A_{\alpha }\subseteq \sum _{\beta \in S,\beta <\alpha }{A}_{\beta }.$$

$$S=\Biggl{\{}\sum_{\alpha\in J_{\beta}}a_{\alpha\beta}x_{\alpha}=~{}0\mid\beta<\lambda,J_{\beta}\in[\lambda]^{<\omega},(\forall\alpha\in J_{\beta})(a_{\alpha\beta}\in\mathbb{Z}),\sum_{\alpha\in J_{\beta}}a_{\alpha\beta}e_{\alpha}=k_{\beta}\Biggr{\}}$$

Full text

This work has been supported by Charles Univ. Research Centre program No. UNCE/SCI/022 and by grant GAČR 17-23112S

1. Introduction and preliminaries

In what follows, group means always an abelian group, i.e. a $\mathbb{Z}$-module. We say that a system $S$ of homogeneous $\mathbb{Z}$-linear equations with a set $X=\{x_{i}\mid i\in I\}$ of variables is nontrivially solvable in a group $H$ if there exists a mapping $f:X\to H\setminus\{0\}$ such that, whenever $\sum_{j\in J}a_{j}x_{j}=0$ is an equation from $S$ (where $J$ is a finite subset of $I$ and $a_{j}\in\mathbb{Z}$ for each $j\in J$), then $\sum_{j\in J}a_{j}f(x_{j})=0$ holds in $H$.

This notion of nontriviality is a little bit unusual. If we assume instead that the mapping $f$ goes to $H$ and it is not constantly zero on all $x\in X$ that appear in the system $S$, we say that the system $S$ is weakly nontrivially solvable in $H$. More natural as it might me, this weaker notion has got one significant disadvantage: unlike with nontrivial solvability, if a system $S$ is weakly nontrivially solvable and $T$ is a nonempty subsystem of $S$, then $T$ need not be weakly nontrivially solvable. Notice also that an empty system $S$ is (weakly) nontrivially solvable by definition.

Motivated by the work in [7], our aim is to characterize the class $\mathcal{S}$ ($\mathcal{WS}$, resp.) of all infinite cardinals $\kappa$ such that any system $S$ of homogeneous $\mathbb{Z}$-linear equations is nontrivially (weakly nontrivially, resp.) solvable in $\mathbb{Z}$ provided that each subsystem $T\subseteq S$ of cardinality $<\kappa$ is nontrivially (weakly nontrivially, resp.) solvable in $\mathbb{Z}$. In [7, Section 2.2], the authors present several well-known examples of countable $S$ which show in ZF that $\aleph_{0}\not\in\mathcal{S}\cup\mathcal{WS}$. They also discuss various interesting related questions in ZF: among other things, they provide a model of ZF without choice where $\aleph_{1}\not\in\mathcal{S}$ while they note that the result is not known in ZFC.

In this short note, we use $\kappa$-free groups with trivial dual to show that ZFC actually proves $\aleph_{n}\not\in\mathcal{S}$ for each $n\in\omega$. Moreover, it is consistent with ZFC that $\mathcal{S}=\mathcal{WS}=\varnothing$. On the other hand, we are able to prove that $\kappa\in\mathcal{WS}\cap\mathcal{S}$ whenever there exits a regular $\mathcal{L}_{\omega_{1}\omega}$-compact cardinal below $\kappa$ (see Corollary 2.2 and Theorem 3.2). A detailed discussion is contained in Section 4.

For an unexplained terminology, we recommend, for instance, the very well-written extensive book [4].

2. The case of $\mathcal{S}$

Recall that, given an uncountable cardinal $\nu$, we say that a cardinal $\kappa$ is $\mathcal{L}_{\nu\omega}$-compact if every $\kappa$-complete filter on any set $I$ can be extended to a $\nu$-complete ultrafilter. Observe that a cardinal $\mu$ is $\mathcal{L}_{\nu\omega}$-compact whenever there exists an $\mathcal{L}_{\nu\omega}$-compact cardinal $\lambda$ such that $\lambda\leq\mu$. This is obviously a large cardinal notion for the existence of an $\mathcal{L}_{\nu\omega}$-compact cardinal implies the existence of a measurable cardinal.

The following result is a classic one. We give a proof for the reader’s convenience.

Proposition 2.1**.**

Let $\lambda$ be a regular $\mathcal{L}_{\nu\omega}$-compact cardinal and $\mathcal{Z}$ a structure in a first order language $L$ with $|Z|<\nu$. Then a system $S$ consisting of first order formulas in variables from a set $X$ is realized in $\mathcal{Z}$ provided that each its subsystem $T$ of cardinality $<\lambda$ is realized in $\mathcal{Z}$.

Proof.

First, let $E$ denote the set $Z^{X}$ of all mappings from $X$ to $Z$. By the assumption, for each $T\in[S]^{<\lambda}$, there exists $e\in E$ such that $\mathcal{Z}\models\varphi[e]$ for each $\varphi\in T$. Let $\mathcal{F}$ be the filter on $E$ generated by the sets $E_{T}=\{e\in E\mid\mathcal{Z}\models\varphi[e]\hbox{ for all }\varphi\in T\}$. Since $\lambda$ is regular, we see that $\mathcal{F}$ is a $\lambda$-complete filter. Let $\mathcal{G}$ denote an extension of $\mathcal{F}$ to a $\nu$-complete ultrafilter.

For each $(x,z)\in X\times Z$, put $E_{x,z}=\{e\in E\mid e(x)=z\}$ and define $f\in Z^{X}$ by the assignment $f(x)=z\Leftrightarrow E_{x,z}\in\mathcal{G}$. We can do it like that since the ultrafilter $\mathcal{G}$ picks, for each fixed $x\in X$, exactly one element from the disjoint partition $E=\bigcup_{z\in Z}E_{x,z}$; recall that $|Z|<\nu$.

Now let $\varphi\in S$ be arbitrary and $x_{1},\dots,x_{n}$ be variables freely occuring in $\varphi$. Then $\varnothing\neq E_{\{\varphi\}}\cap\bigcap_{i=1}^{n}E_{x_{i},f(x_{i})}\in\mathcal{G}$, and so $f\in E_{\{\varphi\}}$. We conclude that $S$ is realized in $\mathcal{Z}$ using the evaluation $f$. ∎

Corollary 2.2**.**

Let $\kappa$ be a cardinal and $\lambda\leq\kappa$ a regular $\mathcal{L}_{\omega_{1}\omega}$-cardinal. Then every nonempty system $S$ of homogeneous $\mathbb{Z}$-linear equations in variables from a set $X$ is nontrivially solvable in $\mathbb{Z}$ whenever each its nonempty subsystem of cardinality $<\kappa$ is nontrivially solvable in $\mathbb{Z}$. In other words $\kappa\in\mathcal{S}$.

Proof.

In the system $S$ replace each equation $\psi$ in variables $x_{1},\dots,x_{n}\in X$ by the formula $\psi\,\&\bigwedge_{i=1}^{n}x_{i}\neq 0$ and use Proposition 2.1. ∎

Before we turn our attention to the negative part, we need one preparatory lemma which holds in the general context of $R$-modules over a (commutative) noetherian domain. Recall, that, for a module $M\in\hbox{{\rm Mod-}}R$ and an ordinal number $\sigma$, an increasing chain $\mathcal{M}=(M_{\alpha}\mid\alpha\leq\sigma)$ of submodules of $M$ is called a filtration of $M$ if $M_{0}=0,M_{\beta}=\bigcup_{\alpha<\beta}M_{\alpha}$ whenever $\beta\leq\sigma$ is a limit ordinal, and $M_{\sigma}=M$.

Lemma 2.3**.**

Let $R$ be an infinite noetherian domain, $M$ a free $R$-module of rank $\mu\geq\aleph_{0}$, and $\mathcal{M}=(M_{\alpha}\mid\alpha\leq\sigma)$ be a filtration of $M$ where, for all $\alpha<\sigma$, $M_{\alpha+1}=M_{\alpha}+\langle a_{\alpha}\rangle$ with $a_{\alpha}\in M\setminus M_{\alpha}$. For each $\alpha<\sigma$, let $z_{\alpha}\in R$ be arbitrary.

Then there is a homomorphism $\psi:M\to R$ such that $\psi(a_{\alpha})\neq z_{\alpha}$ for all $\alpha<\sigma$.

Proof.

First, assume that $\mu=\aleph_{0}$. Let $\{g_{n}\mid n<\omega\}$ be a set of free generators of $M$. For each $\alpha<\sigma$, we express $a_{\alpha}$ as $\sum_{n\in I_{\alpha}}b_{n\alpha}g_{n}$, where $I_{\alpha}$ is a finite subset of $\omega$ and $b_{n\alpha}\in R\setminus\{0\}$ for every $n\in I_{\alpha}$.

Using the fact that a free $R$-module of finite rank is noetherian, we infer that, for each $n<\omega$, the set $A_{n}=\{\alpha<\sigma\mid I_{\alpha}\subseteq\{0,1,\dots,n\}\}$ is finite. Note that $\sigma=\bigcup_{n<\omega}A_{n}$. On the free generators of $M$, we recursively construct a homomorphism $\psi:M\to R$ as follows:

Let $\psi(g_{0})$ be arbitrary such that, for each $\alpha\in A_{0}$, $b_{0\alpha}\psi(g_{0})\neq z_{\alpha}$. There is always an applicable choice by the hypothesis on $R$. Assume that $n>0$, $\psi(g_{n-1})$ is defined, and $\psi(a_{\alpha})\neq z_{\alpha}$ for each $\alpha\in A_{n-1}$.

We define $\psi(g_{n})$ arbitrarily in such a way that, for each $\alpha\in A_{n}\setminus A_{n-1}$, we have

[TABLE]

This is possible, since $A_{n}\setminus A_{n-1}$ is finite, $b_{n\alpha}\neq 0$ for each $\alpha$ from this set, and $R$ is an infinite domain. It immediately follows that $\psi(a_{\alpha})\neq z_{\alpha}$ for each $\alpha\in A_{n}$.

Now, let $\mu$ be an uncountable cardinal. Again, let $\{g_{\beta}\mid\beta<\mu\}$ be a set of free generators of $M$, and put $G_{B}=\langle g_{\beta}\mid\beta\in B\rangle$ for all $B\subseteq\mu$.

We use ideas from [6, Section 7.1]. First, we set $A_{\alpha}=\langle a_{\alpha}\rangle\leq M$. We say that a subset $S$ of the ordinal $\sigma$ is ‘closed’ if every $\alpha\in S$ satisfies

[TABLE]

Notice that any ordinal $\alpha\leq\sigma$ is a ‘closed’ subset of $\sigma$. For a ‘closed’ subset $S$, we define $M(S)=\sum_{\alpha\in S}A_{\alpha}$. The results from [6, Section 7.1] give us the following:

1. (1)

For a system $(S_{i}\mid i\in I)$ of ‘closed’ subsets, $\bigcap_{i\in I}S_{i}$ and $\bigcup_{i\in I}S_{i}$ is ‘closed’ as well. 2. (2)

For $S,S^{\prime}$ ‘closed’ subsets of $\sigma$, we have $S\subseteq S^{\prime}\Longleftrightarrow M(S)\subseteq M(S^{\prime})$. 3. (3)

Let $S$ be a ‘closed’ subset of $\sigma$ and $X$ be a countable subset of $M$. Then there is a ‘closed’ subset $S^{\prime}$ such that $M(S)\cup X\subseteq M(S^{\prime})$ and $|S^{\prime}\setminus S|<\aleph_{1}$.

Using the properties listed above, we are going to construct a filtration $\mathcal{N}=(M(S_{\alpha})\mid\alpha\leq\mu)$ of $M$ such that, for each $\alpha<\mu$, a) $S_{\alpha}$ is ‘closed’, b) $S_{\alpha+1}\setminus S_{\alpha}$ is countable, and c) there exists $B_{\alpha}\subseteq\mu$ such that $G_{B_{\alpha}}=M(S_{\alpha})$ and $\alpha\subseteq B_{\alpha}$.

We proceed by the transfinite recursion, starting with $S_{0}=B_{0}=\varnothing$. Let $S_{\alpha}$ and $B_{\alpha}$ be defined and $\alpha<\mu$. Then $|S_{\alpha}|+|B_{\alpha}|<\mu$ (using b) and c)). Let $B^{0}\supseteq B_{\alpha}\cup\{\alpha\}$ be any subset of $\mu$ with $|B^{0}\setminus B_{\alpha}|=\aleph_{0}$. By $(3)$, we find $S^{0}\supseteq S_{\alpha}$ such that $M(S^{0})\supseteq G_{B^{0}}$ and $|S^{0}\setminus S_{\alpha}|<\aleph_{1}$. Assuming $B^{n},S^{n}$ are defined for $n<\omega$, we can find $B^{n+1}\supseteq B^{n}$ with $|B^{n+1}\setminus B^{n}|<\aleph_{1}$ such that $G_{B^{n+1}}\supseteq M(S^{n})$, and $S^{n+1}\supseteq S^{n}$ with $|S^{n+1}\setminus S^{n}|<\aleph_{1}$ such that $M(S^{n+1})\supseteq G_{B^{n+1}}$. Put $S_{\alpha+1}=\bigcup_{n<\omega}S^{n}$ and $B_{\alpha+1}=\bigcup_{n<\omega}B^{n}$. This completes the isolated step. In limit steps, we simply take unions. Since $M(S_{\mu})=M$, we have $S_{\mu}=\sigma$ by $(2)$.

Now, for each $\alpha<\mu$, we have the countable sets $C_{\alpha}=B_{\alpha+1}\setminus B_{\alpha}$ and $T_{\alpha}=S_{\alpha+1}\setminus S_{\alpha}$, and the canonical projection $\pi_{\alpha}:M(S_{\alpha+1})\to G_{C_{\alpha}}$. Let $\tau$ be the ordinal type of $(T_{\alpha},<)$, and fix an order-preserving bijection $i:\tau\to T_{\alpha}$.

Since $S_{\alpha}\cup(S_{\alpha+1}\cap\beta)$ is ‘closed’ for any $\beta\leq\sigma$ by $(1)$, the part $(2)$ yields that the chain $(N_{\beta}\mid\beta\leq\tau)$ of modules defined as $N_{\beta}=M(S_{\alpha}\cup(S_{\alpha+1}\cap i(\beta)))$, for $\beta<\tau$, and $N_{\tau}=M(S_{\alpha+1})$ is strictly increasing. Notice that $N_{0}=M(S_{\alpha})$.

If we put $\bar{N}_{\beta}=\pi_{\alpha}[N_{\beta}]$ for all $\beta\leq\tau$, it follows that the strictly increasing chain $(\bar{N}_{\beta}\mid\beta\leq\tau)$ is a filtration of the free module $G_{C_{\alpha}}$ of countable rank. Moreover, for each $\beta<\tau$, we have $\bar{N}_{\beta+1}=\bar{N}_{\beta}+\langle\pi_{\alpha}(a_{i(\beta)})\rangle$.

Finally, we recursively define the homomorphism $\psi:M\to R$. Let $\alpha<\mu$ and assume that $\psi\restriction G_{B_{\alpha}}$ is constructed with the property $\psi(a_{\gamma})\neq z_{\gamma}$ for all $\gamma\in S_{\alpha}$. By the already proven part for $\mu=\aleph_{0}$, we can define $\psi\restriction G_{C_{\alpha}}$ in such a way that $\psi(\pi_{\alpha}(a_{\gamma}))\neq z_{\gamma}-\psi(a_{\gamma}-\pi_{\alpha}(a_{\gamma}))$ for all $\gamma\in T_{\alpha}$; observe that the right-hand side of the inequality is already defined since $a_{\gamma}-\pi_{\alpha}(a_{\gamma})\in G_{B_{\alpha}}$. We immediately get $\psi(a_{\gamma})\neq z_{\gamma}$ for all $\gamma\in S_{\alpha+1}$.

∎

For the negative part, we start with an uncountable cardinal $\kappa$ and a $\kappa$-free group $G$ with the trivial dual, i.e. with the property $G^{*}:=\operatorname{Hom}(G,\mathbb{Z})=0$; here, $\kappa$-free means that any $<\!\kappa$-generated subgroup of $G$ is free. We will discuss the existence of such groups, as well as the question whether $G$ can be taken with $|G|=\kappa$, later on. Firstly, we show how the existence of such $G$ implies that $\kappa\not\in\mathcal{S}$.

Let us denote by $\lambda$ the cardinality of $G$ and express $G$ as a quotient $F/K$ where $F$ is a free group of rank $\lambda$. Notice that $\lambda\geq\kappa$. Let $\pi:F\to F/K$ denote the canonical projection and let $\{e_{\alpha}\mid\alpha<\lambda\}$ be a set of free generators of the group $F$. For each $A\subseteq\lambda$, let $F_{A}$ denote the subgroup of $F$ generated by $\{e_{\alpha}\mid\alpha\in A\}$. We can w.l.o.g. assume that

[TABLE]

The group $K$ is also free of rank $\lambda$. If it had a smaller rank, $G$ would have possessed a free direct summand—a contradiction with $G^{*}=0$. Let $\{k_{\beta}\mid\beta<\lambda\}$ denote a set of (free) generators of the group $K$. Consider the uncountable set

[TABLE]

of homogeneous $\mathbb{Z}$-linear equations with the set $\{x_{\alpha}\mid\alpha<\lambda\}$ of variables. We will show that this is the desired counterexample.

First of all, $S$ does not have even a weakly nontrivial solution in $\mathbb{Z}$. Indeed, any such solution would define a nonzero homomorphism $\psi$ from $F$ to $\mathbb{Z}$ which is zero on $K$. Hence $\psi$ would provide for a nonzero homomorphism from $G$ to $\mathbb{Z}$, a contradiction.

On the other hand, we can show

Proposition 2.4**.**

Any system $T\subseteq S$ with cardinality $<\kappa$ has a nontrivial solution in $\mathbb{Z}$.

Proof.

Let $A\in[\lambda]^{<\kappa}$ be an infinite set such that whenever $x_{\alpha}$ appears in an equation from $T$ then $\alpha\in A$. Put $M=\operatorname{Im}(\pi\restriction F_{A})$.

Since $G$ is $\kappa$-free, $M$ is a free group (of infinite rank). Let $\sigma$ denote the ordinal type of $A$ and fix an order-preserving bijection $i:\sigma\to A$. For each $\alpha\leq\sigma$, set $M_{\alpha}=\langle\pi(e_{i(\beta)})\mid\beta<\alpha\rangle$. Then $(M_{\alpha}\mid\alpha\leq\sigma)$ is a filtration of $M$ such that $M_{\alpha+1}=M_{\alpha}+\langle\pi(e_{i(\alpha)})\rangle$ where $\pi(e_{i(\alpha)})\not\in M_{\alpha}$ for all $\alpha<\sigma$ (using $(*)$).

Applying Lemma 2.3 with $R=\mathbb{Z}$ and $z_{\gamma}=0$ for all $\gamma<\sigma$, we obtain a homomorphism $\psi:M\to\mathbb{Z}$ such that $\psi(\pi(e_{\alpha}))\neq 0$ for all $\alpha\in A$. The assignment $x_{\alpha}\mapsto\psi(\pi(e_{\alpha}))$, $\alpha\in A$, is the desired nontrivial solution of the system $T$ in $\mathbb{Z}$. ∎

Corollary 2.5**.**

Let $\kappa$ be an uncountable cardinal. If there exists a $\kappa$-free group $G$ with $G^{*}=0$, then $\kappa\not\in\mathcal{S}\cup\mathcal{WS}$.

3. The case of $\mathcal{WS}$

For the weaker notion of nontrivial solvability, we have the following general result.

Proposition 3.1**.**

Let $\kappa$ be an uncountable cardinal. The following conditions are equivalent:

1. (1)

There exists a regular cardinal $\lambda\leq\kappa$ which is $\mathcal{L}_{\omega_{1}\omega}$-compact. 2. (2)

There is a regular cardinal $\lambda\leq\kappa$ such that each group $A\in\operatorname{Ker}\operatorname{Hom}(-,\mathbb{Z})$ is the sum of its subgroups of cardinality $<\lambda$ which are contained in $\operatorname{Ker}\operatorname{Hom}(-,\mathbb{Z})$. 3. (3)

For any nonempty system $S$ of homogeneous $\mathbb{Z}$-linear equations such that $S$ has no weakly nontrivial solution in $\mathbb{Z}$, and any $C\in[S]^{<\kappa}$, there exists $T\in[S]^{<\kappa}$ such that $C\subseteq T$ and $T$ has no weakly nontrivial solution in $\mathbb{Z}$.

Proof.

The equivalence of $(1)$ and $(2)$ follows directly from [1, Corollary 5.4]. Let us show that $(2)$ is equivalent to $(3)$. To this end, we are going to use the following two-way translation.

Given any system $S=\{k_{j}=0\mid j\in J\}$ of homogeneous $\mathbb{Z}$-linear equations with the set $X$ of variables, we can build a group $A=F/K$ where $F$ is freely generated by the elements of the set $X$ and $K$ is generated by the set $\{k_{j}\mid j\in J\}$. Then $\operatorname{Hom}(A,\mathbb{Z})=0$ if and only if $S$ has no weakly nontrivial solution in $\mathbb{Z}$. On the other hand, for a given group $A$ and its presentation $F/K$ where $F$ is freely generated by a set $X$, the same equivalence holds for the system $S=\{k_{j}=0\mid j\in J\}$ of homogeneous $\mathbb{Z}$-linear equations where $\{k_{j}\mid j\in J\}$ is a fixed set of generators of $K$ expressed as $\mathbb{Z}$-linear combinations of elements from the set $X$.

Proving $(2)\Longrightarrow(3)$, we start with a system $S$ and a set $C\in[J]^{<\kappa}$. Consider the group $A$ constructed for $S$ as in the previous paragraph, and let $Y_{0}$ denote the set of all the elements from $X$ appearing in equations $k_{j}=0,j\in C$.

Let $\mu\geq\lambda$ be a regular uncountable cardinal such that $|C|<\mu\leq\kappa$. Since $\operatorname{Ker}\operatorname{Hom}(-,\mathbb{Z})$ is closed under direct sums and quotients, and $\mu$ is regular, there exists, by $(2)$, $G_{0}\in\operatorname{Ker}\operatorname{Hom}(-,\mathbb{Z})$ such that $G_{0}$ is a subgroup of $A$, $|G_{0}|<\mu$ and $Y_{0}+K:=\{y+K\mid y\in Y_{0}\}\subseteq G_{0}$. Now, take any $Y_{1}\in[X]^{<\mu},Y_{0}\subseteq Y_{1}$ such that:

1. $(a)$

$G_{0}$ is contained in the subgroup of $A$ generated by $Y_{1}+K$. 2. $(b)$

There exists $C_{0}\in[J]^{<\mu}$ such that $\langle Y_{0}\rangle\cap K$ is contained in the subgroup of $K$ generated by $\{k_{j}\mid j\in C_{0}\}$, and $Y_{1}$ contains all the elements from $X$ appearing in equations $k_{j}=0,j\in C_{0}$.

For this $Y_{1}$, we obtain, using $(2)$, a subgroup $G_{1}$ of $A$ with $|G_{1}|<\mu$, and so on.

After $\omega$ steps, we have the group $G=\sum_{n<\omega}G_{n}\in\operatorname{Ker}\operatorname{Hom}(-,\mathbb{Z})$ generated by $Y+K$ where $Y=\bigcup_{n<\omega}Y_{n}\in[X]^{<\mu}$. By the construction, we have also $G=\langle y+K\mid y\in Y\rangle\cong\langle Y\rangle/\langle k_{j}\mid j\in\bigcup_{n<\omega}C_{n}\rangle$. Finally, we put $T=\{k_{j}=0\mid j\in\bigcup_{n<\omega}C_{n}\}$.

Now, let us prove the implication $\neg(1)\Longrightarrow\neg(3)$. First, assume that $\kappa$ is not $\mathcal{L}_{\omega_{1}\omega}$-compact. Following [1, Theorem 5.3] and its proof, we start with $A=\mathbb{Z}^{I}/\mathcal{F}$ where $\mathcal{F}$ is a $\kappa$-complete filter on $I$ which cannot be extended to an $\omega_{1}$-complete ultrafilter. From the latter part, it follows that $\operatorname{Hom}(A,\mathbb{Z})=0$. The $\kappa$-completeness of $\mathcal{F}$, on the other hand, assures that any subgroup of $A$ of cardinality $<\kappa$ can be embedded into $\mathbb{Z}^{I}$.

Consider a system $S$ of homogeneous $\mathbb{Z}$-linear equations associated to the group $A$ presented as $F/K$ where $F$ is freely generated by a set $X$. We can w. l. o. g. assume that no $x\in X$ is contained in $K$. Let $C\in[J]^{<\kappa}$ be non-empty. We shall show that the system $\{k_{j}=0\mid j\in C\}$ has weakly nontrivial solution in $\mathbb{Z}$.

As in the proof of the other implication, we can possibly enlarge $C$ to some $D\subseteq J$ such that $|D|\leq|C|+\aleph_{0}$ and—denoting by $Y$ the set of all the elements from $X$ appearing in equations $k_{j}=0,j\in D$—$\,\langle y+K\mid y\in Y\rangle\cong\langle Y\rangle/\langle k_{j}\mid j\in D\rangle$. Let us denote the latter group by $H$ and fix an embedding $i:H\to\mathbb{Z}^{I}$ (which exists since $|H|<\kappa$).

Let $y\in Y$ be any element appearing in (one of the) equations $k_{j}=0,j\in C$. Since $i(y+K)\neq 0$ there is a projection $\pi:\mathbb{Z}^{I}\to\mathbb{Z}$ such that $\pi i(y+K)\neq 0$. The assignment $x\mapsto\pi i(x+K)$ defines the desired weakly nontrivial solution of the system $\{k_{j}=0\mid j\in C\}$ in $\mathbb{Z}$.

It remains to tackle the possibility that $\kappa$ is the least $\mathcal{L}_{\omega_{1}\omega}$-compact cardinal and $\kappa$ is singular. We know by [2] that $\gamma=cf(\kappa)$ is greater than or equal to the first measurable cardinal in this case. Let $(\kappa_{\alpha}\mid\alpha<\gamma)$ be an increasing sequence of cardinals $<\kappa$ converging to $\kappa$.

Consider the group $A=\bigoplus_{\alpha<\gamma}A_{\alpha}$ where, for each $\alpha<\gamma$, $A_{\alpha}\in\operatorname{Ker}\operatorname{Hom}(-,\mathbb{Z})$ is not a sum of its subgroups of cardinality $<\kappa_{\alpha}$ which belong to $\operatorname{Ker}\operatorname{Hom}(-,\mathbb{Z})$. Assume, for the sake of contradiction, that $(3)$ holds for the system $S$ of homogeneous $\mathbb{Z}$-linear equations associated to the group $A$ (more precisely, to its presentation $F/K$).

By the definition of $A$, there exists, for each $\alpha<\gamma$, an element $a_{\alpha}\in A$ such that $a_{\alpha}$ is not contained in any subgroup $H$ of $A$ of cardinality $<\kappa_{\alpha}$ with the property $\operatorname{Hom}(H,\mathbb{Z})=0$.

We know that there is $C_{0}\in[J]^{<\kappa}$ and $Y_{0}\subseteq X$ consisting of the elements from $X$ appearing in the equations $k_{j}=0,j\in C_{0}$ such that $\{a_{\alpha}\mid\alpha<\gamma\}\subseteq\langle y+K\mid y\in Y_{0}\rangle\cong\langle Y_{0}\rangle/\langle k_{j}\mid j\in C_{0}\rangle$.

For this $C_{0}$, we obtain a corresponding $T_{0}\in[J]^{<\kappa}$ using $(3)$. We continue by finding $C_{1}\in[J]^{<\kappa}$ and $Y_{1}\in[X]^{<\kappa}$ such that $T_{0}\subseteq C_{1},Y_{0}\subseteq Y_{1}$ and $\langle y+K\mid y\in Y_{1}\rangle\cong\langle Y_{1}\rangle/\langle k_{j}\mid j\in C_{1}\rangle$, and so forth.

Put $T=\bigcup_{n<\omega}T_{n}=\bigcup_{n<\omega}C_{n}$ and $Y=\bigcup_{n<\omega}Y_{n}$. The system $\{k_{j}=0\mid j\in T\}$ has cardinality $<\kappa$ (since $\gamma$ is uncountable) and it has no weakly nontrivial solution in $\mathbb{Z}$. Whence the subgroup $H=\langle y+K\mid y\in Y\rangle\cong\langle Y\rangle/\langle k_{j}\mid j\in T\rangle$ of $A$ belongs to $\operatorname{Ker}\operatorname{Hom}(-,\mathbb{Z})$. However, this is impossible since $a_{\alpha}\in H$ for $\alpha<\gamma$ satisfying $|H|<\kappa_{\alpha}$. ∎

In the proof above, we have actually showed a little bit more. In fact, we have the following

Theorem 3.2**.**

Let $\kappa$ be a cardinal, and assume that $\kappa$ is not at the same time singular and the least $\mathcal{L}_{\omega_{1}\omega}$-compact cardinal. The following conditions are equivalent:

1. (1)

$\kappa$* is $\mathcal{L}_{\omega_{1}\omega}$-compact.* 2. (2)

Any nonempty system $S$ of homogeneous $\mathbb{Z}$-linear equations has a weakly nontrivial solution in $\mathbb{Z}$ provided that each its nonempty subsystem of cardinality $<\kappa$ has one. In other words, $\kappa\in\mathcal{WS}$.

Proof.

The implication ‘$(1)\Longrightarrow(2)$’ follows immediately from ‘$(1)\Longrightarrow(3)$’ in Proposition 3.1. The other implication then from the first part of the proof of $`\neg(1)\Longrightarrow\neg(3)$’ in Proposition 3.1.

∎

4. Concluding remarks

The problem of existence of $\kappa$-free groups with trivial dual turns out to be rather delicate. Under the assumption V = L (even a much weaker one), there are $\kappa$-free groups with trivial dual for any uncountable cardinal $\kappa$. Moreover, if $\kappa$ is regular and not weakly compact, then the groups can be constructed of cardinality $\kappa$, see [3]. If $\kappa$ is singular or weakly compact, then $\kappa$-free implies $\kappa^{+}$-free. For more information on the topic, we refer to [4, Chapter VII]. Anyway, we have $\mathcal{S}=\mathcal{WS}=\varnothing$ under V = L by Corollary 2.5.

In [5], Göbel and Shelah show in ZFC that $\aleph_{n}$-free groups with cardinality $\beth_{n}$ and trivial dual exist for all $0<n<\omega$. This is further generalized in [8]111Very heavy in content. Unpublished outside arXiv.org so far., where Shelah proves in ZFC the existence of $\kappa$-free groups with trivial dual for any uncountable $\kappa<\aleph_{\omega_{1}\cdot\omega}$. On the other hand, he also shows (modulo the existence of a supercompact cardinal) that it is relatively consistent with ZFC that there is no $\aleph_{\omega_{1}\cdot\omega}$-free group with trivial dual.

By Corollary 2.5, we thus know in ZFC that $\kappa\not\in\mathcal{S}$ for $\kappa<\aleph_{\omega_{1}\cdot\omega}$. However, we do not know what happens for larger cardinals $\kappa$ since the existence of a $\kappa$-free group with trivial dual is just a sufficient condition for $\kappa\not\in\mathcal{S}$. We have only the upper bound given by Corollary 2.2. It might still be possible that $\mathcal{S}=\mathcal{WS}$ where for the latter class, we have a decent description in Theorem 3.2. As shown in [1], relative to the existence of a supercompact cardinal, there are models of ZFC where the smallest $\mathcal{L}_{\omega_{1}\omega}$-compact cardinal $\kappa$ is singular. In this only case, we cannot resolve the question whether $\kappa\in\mathcal{WS}$ although we conjecture that this is not the case, which would readily imply that at least $\mathcal{WS}\subseteq\mathcal{S}$ always holds.

A possible direction for further research is to investigate further what more can be proved in ZFC about the class $\mathcal{S}$.

Acknowledgement: I would like to thank Petr Glivický for a fruitful discussion about compactness problems in set theory.