# Countably compact groups having minimal infinite powers

Dikran Dikranjan  Dipartimento di Scienze Matematiche, Informatiche e Fisiche
Università di Udine<EMAIL_ADDRESS>and  Vladimir Uspenskij
Department of Mathematics, Ohio University, Athens, OH 45701, USA
<EMAIL_ADDRESS>
###### Abstract.

We answer the question, raised more than thirty years ago in [10, 12], on
whether the power $G^{\omega}$ of a countably compact minimal Abelian group
$G$ is minimal, by showing that the negative answer is equivalent to the
existence of measurable cardinals. The proof is carried out in the larger
class of sequentially complete groups. We characterize the sequentially
complete minimal Abelian groups $G$ such that $G^{\omega}$ is minimal – these
are exactly those $G$ that contain the connected component of their
completion. This naturally leads to the next step, namely, a better
understanding the structure of the sequentially complete minimal Abelian
groups, and in particular, their connected components which turns out to
depend of the existence of Ulam measurable cardinals. More specifically, all
connected sequentially complete minimal Abelian groups are compact, if Ulam
measurable cardinals do not exist. On the other hand, for every Ulam
measurable cardinal $\sigma$ we build a non-compact torsion-free connected
minimal $\omega$-bounded Abelian group of weight $\sigma$, thereby showing
that the Ulam measurable cardinals are precisely the weights of non-compact
sequentially complete connected minimal Abelian groups.

###### 2020 Mathematics Subject Classification:

Primary 22A05.
Secondary 54H11, 22B05, 54A35, 54B30, 54D25, 54D30, 54H13.
Key words and phrases: minimal group, precompact group, sequentially complete
group, countably compact group, $\omega$-bounded group, compact group,
connected group, totally disconnected group, measurable cardinal

##  1\. Introduction

We denote by ${{\mathbb{N}}}$ set of naturals $\\{0,1,\dots\\}$ (and
${\mathbb{N}}^{*}={\mathbb{N}}\setminus\\{0\\}=\\{1,2,\dots\\}$), by
${{\mathbb{P}}}$ the set all primes, by ${{\mathbb{Z}}}$ the integers, by
${{\mathbb{Q}}}$ the rationals, by ${{\mathbb{R}}}$ the reals, by
${{\mathbb{T}}}$ the unit circle group in ${{\mathbb{C}}}$, by
${\mathbb{Z}}_{p}$ the $p$-adic integers ($p\in{\mathbb{P}}$), by
${\mathbb{Z}}(n)$ the cyclic group of order $n$ ($n\in{\mathbb{N}}^{*}$). The
cardinality of continuum $2^{\omega}$ will be denoted also by
${\mathfrak{c}}$.

All groups considered in this paper (except in §3) will be Abelian, so
additive notation will be always used. We denote by $\widetilde{G}$ the
completion of a topological group $G$ (we do not have to worry about left,
right, and two-sided uniformities, since our groups are Abelian), while $c(G)$
denotes the connected component of the neutral element of a group $G$ (for
brevity, we call $c(G)$ the connected component of $G$).

###  1.1. Minimal groups and their products

Following Stephenson [27], call a Hausdorff topological group $(G,\tau)$
minimal if the topology $\tau$ is a minimal element of the partially ordered,
with respect to inclusion, set of Hausdorff group topologies on $G$. Compact
groups are obviously minimal, the first examples of non-compact minimal groups
were given in [27, 17], while the first examples of non-compact countably
compact minimal groups were given in [3] (see also [10]–[13]).

The problem of preservation of minimality under products (the counterpart of
Tychonoff product theorem for compactness) was raised by G. Choquet at the ICM
in Nice in 1970. The first counter-example was given in [17]. Since then this
problem became a central one in the framework of minimal groups [17, 5, 8, 9,
12, 15, 18, 28, 29]. A general criterion for minimality of arbitrary products
of minimal Abelian groups was produced in [5, 9] (see also [9, Theorem 2.4]
and [8, Theorem 6.2.3]). The minimality of arbitrary powers of an Abelian
group $G$ was characterized earlier by Stoyanov [29], who proved that it is
equivalent to the minimality of $G^{\mathfrak{c}}$ (see Theorem 2.5 for a more
detailed description). Since minimality is preserved by taking direct
summands, the minimality of some power $G^{\kappa}$ implies the minimality of
all smaller powers.

It is natural to expect that adding some additional compact-like properties
may improve productivity of minimal groups. Indeed, all finite powers of a
countably compact minimal group are minimal [12]. On the other hand, there
exist pseudocompact minimal Abelian groups $G$ with $G^{\omega_{1}}$ non-
minimal, but $G^{\omega}$ minimal (for more detail see [6, 12]).

The choice of compact-like property to match with minimality in this paper is
sequential completeness, since it simultaneously generalizes two fundamental
properties like countable compactness and completeness. A topological group
$G$ is sequentially complete if $G$ is sequentially closed in $\widetilde{G}$
(or, equivalently, when every Cauchy sequence in $G$ converges) [13, 14]. We
prove that all finite powers of a sequentially complete minimal group are
minimal (this is a corollary of a more general fact, see Lemma 2.7). This
suggests the following question, set in [10, Question 9] and [12, Question
1.10] in the case of countably compact groups which still remains open since
then:

######  Question 1.1.

Let $G$ be a sequentially complete minimal Abelian group. Is $G^{\omega}$
minimal ?

We shall see below that this question, as well as the question in [12]
regarding countably compact groups, cannot be answered in ZFC.

In order to answer Question 1.1, we characterize first the sequentially
complete Abelian groups $G$ such that $G^{\omega}$ is minimal (i.e., those
providing a positive answer to Question 1.1):

Theorem A. For a sequentially complete minimal Abelian group $G$ the following
are equivalent:

  * (a)

all powers of $G$ are minimal;

  * (b)

$G^{\omega}$ is minimal.

  * (c)

$G$ contains $c(\widetilde{G})$; hence, $c(G)=c(\widetilde{G})$ is compact.

The proof of this theorem is given in §2.2, where we prove it first in the
totally disconnected case, when $G^{\omega}$ is always minimal (Corollary
2.14), i.e., Question 1.1 has positive answer for totally disconnected $G$.
The equivalence of (b) and a weaker form of (c) was established in [15, Main
Theorem] in the much smaller class of $\omega$-bounded minimal Abelian groups.

On the other hand, we shall see in Corollary 1.6 that the assertion “all
powers of a countably compact minimal Abelian groups are minimal” is
equivalent to the non-existence of Ulam-measurable cardinals. Let us recall
that a cardinal $|X|$ is Ulam-measurable (resp., measurable) if there exists
an $\omega_{1}$-complete (resp., $|X|$-complete) free ultrafilter on $X$ (a
filter on a set $X$ is $\kappa$-complete if it is closed under intersections
of families of cardinality $<\kappa$). A cardinal is Ulam-measurable if and
only if it greater than or equal to the first measurable cardinal. The
assumption that there exist no Ulam-measurable cardinals is known to be
consistent with ZFC, while their existence implies consistency of ZFC and
hence no proof of that existence is to be expected [22].

Uncountable measurable cardinals are Ulam-measurable, while the least Ulam-
measurable cardinal is also measurable.

For the connected case Theorem A gives the following striking consequence
characterizing the minimality of $G^{\omega}$ by compactness of $G$:

######  Corollary 1.2.

A connected sequentially complete minimal Abelian group $G$ is compact if and
only if $G^{\omega}$ is minimal.

This corollary shows that in order to positively answer Question 1.1 in the
connected case, we have to check whether the connected sequentially complete
minimal Abelian group $G$ are compact. This issue turns out to be non–trivial,
we shall face it in §1.2 which mainly deals with this aspect of the structure
of sequentially complete minimal Abelian groups (see Question 1.4).

Here is another application of Theorem A. The general criterion for minimality
of arbitrary products of Abelian groups from [8, Theorem 6.2.3], simplifies in
the case of sequentially complete group (due to Lemma 2.7), so that applying
the criterion from [8, Corollary 6.2.8] one gets the following corollary (for
a proof see §2.2).

######  Corollary 1.3.

Let $\\{G_{i}\\}_{i\in I}$ be a family of sequentially complete Abelian
groups. Then $G=\prod_{i\in I}G_{i}$ is minimal if and only if every countable
subproduct is minimal. In such a case there exists a co-finite subset
$J\subseteq I$ such that all groups $G_{i}^{\omega}$, $i\in J$, are minimal.

Corollary 1.2 implies that if the groups $G_{i}$ in the above theorem are also
connected, then $G$ is minimal if and only if all but finitely many of the
groups $G_{i}$ are compact.

According to [12, Theorem 1.5] the last part of Corollary 1.3 cannot be
inverted: there exists a family $\\{G_{n}:n\in{\mathbb{N}}\\}$ of
$\omega$-bounded Abelian groups such that all powers $G_{n}^{\lambda}$ are
minimal, but $\prod_{n=0}^{\infty}G_{n}$ is not minimal.

###  1.2. The structure of sequentially complete minimal groups

Ulam-measurable cardinals already appeared in the context of comparison
between compactness and countable compactness [1], [4], [32]. Relaxing
countable compactness to sequential completeness and adding minimality to the
mix we consider the following

######  Question 1.4.

Is every connected sequentially complete minimal Abelian group $G$ compact?

Connectedness cannot be omitted here, since there exist minimal
$\omega$-bounded proper dense subgroups of ${\mathbb{Z}}_{p}^{\omega_{1}}$
[11] (they are obviously totally disconnected).

According to Corollary 1.2 a positive answer to Question 1.1 is equivaent to a
positive answer to Question 1.4.

It turns out that the answer to Question 1.4 depends on the existence of
measurable cardinals, hence Question 1.4 and Question 1.1 cannot be answered
in ZFC.

Theorem B. Let $\alpha\geq\omega$ be a cardinal. Then the following conditions
are equivalent:

  * (a)

$\alpha$ is Ulam-measurable;

  * (b)

there exists a non-compact connected, sequentially complete, minimal Abelian
group $G$ of weight $\alpha$;

  * (c)

there exists a non-compact connected, $\omega$-bounded, minimal torsion-free
Abelian group $G$ of weight $\alpha$.

Moreover, if $\alpha$ is the least Ulam-measurable cardinal, then there exists
a connected non-compact minimal torsion-free Abelian group $G$ of weight
$\alpha$ that is $\beta$-bounded for every $\beta<\alpha$.

The proof of this theorem is given in §2.2. Since the least Ulam-measurable
cardinal is a measurable cardinal, we prove there a sightly more general
version of the last assertion of the theorem. Namely, for every measurable
cardinal $\alpha$ there exists a connected non-compact minimal torsion-free
Abelian group $G$ of weight $\alpha$ that is $\beta$-bounded for every
$\beta<\alpha$. It should be noted here that the last property is very close
to compactness, since a space $X$ that is $w(X)$-bounded must be compact.

On the other hand, the implication (b) $\to$ (a) of the above theorem follows
from the fact that a connected sequentially complete minimal Abelian group of
non-measurable size is compact due to item (b) of the following:

Theorem C. Let $G$ be a sequentially complete minimal Abelian group.

  * (a)

If $c(G)$ is compact, then $c(G)=c(\widetilde{G})$ is compact and $G/c(G)$ is
sequentially complete and minimal.

  * (b)

If $w(c(G))$ is not Ulam-measurable, then $c(G)$ is compact.

In particular, a sequentially complete Abelian group $G$ with non Ulam-
measurable $w(c(G))$ is minimal if and only if $c(G)$ is compact and $G/c(G)$
is minimal (the sufficiency uses the three space property, see Proposition
2.8, or [18]). This reduces the study of countably compact minimal Abelian
groups with small connected component to that of totally disconnected groups.

Now we come back to Question 1.1. First we give a sufficient condition for a
positive answer:

######  Corollary 1.5.

Let $G$ be a sequentially complete minimal Abelian group such that $w(c(G))$
is not Ulam measurable. Then $G^{\omega}$ is minimal.

Indeed, the hypotheses of the corollary imply that $c(G)=c(\widetilde{G})$ is
compact (in view of Theorem C(b)), so $G^{\omega}$ is minimal, by Theorem A.
This proves item (a) of the next corollary, in view of Theorem B. Actually,
the next corollary shows that Question 1.1 cannot be answered neither for
countably compact groups, nor for sequentially complete groups.

######  Corollary 1.6.

(a) Under the assumption that there exist no Ulam-measurable cardinals, all
powers of a sequentially complete minimal Abelian group are minimal, i.e.,
Question 1.1 has a positive answer.

(b) Under the assumption that there exist Ulam-measurable cardinals, there
exists an $\omega$-bounded minimal Abelian group $G$ such that $G^{\omega}$ is
not minimal, Question 1.1 has a negaitive answer.

Item (b) follows from item (b) of Theorem B, which ensures the existence of a
non-compact connected, sequentially complete, minimal Abelian group $G$ of
weight $\alpha$, whenever $\alpha$ is an Ulam-measurable cardinal. Then
$G^{\omega}$ is not minimal, according to Corollary 1.2.

Finally, we offer a description of the sequentially complete Abelian group $G$
such that $G^{\omega}$ is minimal, i.e., those satisfying the equivalent
conditions of Theorem A. They turn out to be extensions of a compact group by
a minimal Abelian group that is a direct product of bounded minimal
$p$-groups.

Theorem D. Let $G$ be a sequentially complete Abelian group such that
$G^{\omega}$ is minimal. Then there exists a compact subgroup $N$ of $G$ such
that $G/N\cong\prod_{p\in{\mathbb{P}}}B_{p}$, where $B_{p}$ is a bounded
$p$-torsion minimal group for every prime $p$.

This theorem is proved in §2.2. The subgroup $N$ is “large”, it contains the
compact subgroup $c(\widetilde{G})$ and much more (for a stronger and more
precise form of the theorem see Theorem 2.17).

The paper is organzied as follows. In §2.1 we provide background on minimality
and sequential completeness. In particular, we recall two criteria for
minimality of dense subgroups of compact Abelian groups and we give a
factorization theorem for totally disconnected sequentially complete minimal
Abelian group (Theorem 2.10). Section 2.2 contains the proofs of Theorems A,
B, C, D and Corollary 1.3. A key point of §2.2 is Lemma 2.15, which is the
clue to the proofs of Theorems A, C and D.

### Notation and terminology

Let $G$ be a group and $A$ be a subset of $G$. We denote by $\langle A\rangle$
the subgroup of $G$ generated by $A$. The group $G$ is divisible if for every
$g\in G$ and $n\in{\mathbb{N}}^{*}$ the equation $nx=g$ has a solution in $G$.
For an Abelian group $G$ we set $G[n]=\\{x\in G:nx=0\\}$, for
$n\in{\mathbb{N}}^{*}$, and $Soc(G)=\bigoplus_{p\in{\mathbb{P}}}G[p]$.

We recall some notions of compactness-like conditions in topological groups
and spaces. A Tychonoff space $X$ is pseudocompact if every continuous real-
valued function on $G$ is bounded, and countably compact if every countable
open cover has a finite subcover (equivalently, every sequence in $X$ has a
cluster point). For an infinite cardinal $\alpha$, a group $G$ is
$\alpha$-bounded  if every subset of cardinality $\leq\alpha$ of $G$ is
contained in a compact subgroup of $G$ (some authors use the term
$\omega$-bounded for a different property). Obviously, $\omega$-boundedness
implies countable compactness, and countable compactness implies
pseudocompactness.

For undefined symbols or notions see [8, 19, 20].

##  2\. Proofs of the main results

###  2.1. Background on minimality and sequential completeness

A topological group $G$ is precompact if its completion $\widetilde{G}$ is
compact (or, equivalently, if for any open $U\neq\emptyset$ in $G$ there is a
finite subset $F\subseteq G$ such that $F+U=G$). Peudocompact groups are
precompact. The following fundamental theorem of Prodanov and Stoyanov [25]
(see also [8, Theorem 2.7.7] for an alternative proof) says that the minimal
Abelian groups are precisely the (dense) subgroups of the compact Abelian
groups:

######  Theorem 2.1.

[25] Every minimal Abelian group is precompact.

A subgroup $H$ of a topological Abelian group $G$ is essential if every non-
trivial closed subgroup $N$ of $G$ meets $H$ non-trivially. The following
criterion for minimality of dense subgroups, given by Stephenson [27] and
Prodanov [24], describes the minimal Abelian groups as the dense essential
subgroups of the compact Abelian groups:

######  Theorem 2.2.

Let $G$ be a compact Abelian group and $H$ be a dense subgroup of $G$. Then
$H$ is minimal if and only if $H$ is essential in $G$.

Actually, it is possible to carry out the test of essentiality with only
closed subgroups $N$ of $G$ that are either cyclic $p$-group or copies of
${\mathbb{Z}}$, with $p\in{\mathbb{P}}$ :

######  Lemma 2.3.

[8, Theorem 4.3.7] Let $G$ be a compact Abelian group. A dense subgroup $H$ of
$G$ is essential if and only if $Soc(G)\leq H$ and for every prime $p$ the
subgroup $H$ non-trivially meets every subgroup $N\leq G$ with
$N\cong{\mathbb{Z}}_{p}$.

The reduction in the above lemma is based on a the localization technique
invented by Stoyanov [28] (see also [8, Chapter 4]) that we recall now. For a
prime $p$ an element $x$ of a topological Abelian group $G$ is called
quasi-$p$-torsion if either $x$ is $p$-torsion or $\langle x\rangle$ is
isomorphic to ${\mathbb{Z}}$ equipped with the $p$-adic topology. The set
$td_{p}(G)$ of all quasi-$p$-torsion elements of $G$ is a subgroup of $G$. In
these terms, Lemma 2.3 ensures that $H$ is essential in $G$ if and only if
$td_{p}(H)$ is essential in $td_{p}(G)$ for every $p$ (see [8, Theorem
4.3.7]).

Here we collect some properties of the subgroup $td_{p}(-)$ that will be used
in the sequel, for further properties see [8, Chapter 4],

######  Fact 2.4.

[8]

  * (1)

Let $f:H\to G$ be a continuous homomorphisms of topological Abelian groups.
Then for every prime $p$:

    * (1a)

$f(td_{p}(H))\leq td_{p}(G)$, if $H$ is compact and $f$ is surjective, then
$f(td_{p}(H))=td_{p}(G)$;

    * (1b)

if $f$ is an embedding, then $f(td_{p}(H))=td_{p}(G)\cap f(H)$.

  * (2)

If $\\{G_{i}:i\in I\\}$ is a family of topological Abelian groups, then
$td_{p}(\prod_{i\in I}G_{i})=\prod_{i\in I}td_{p}(G_{i})$.

  * (3)

If $G$ is a totally disconnected compact Abelian group, then for every
$p\in{\mathbb{P}}$ the subgroup $td_{p}(G)$ of $G$ is closed and
$G\cong\prod_{p\in{\mathbb{P}}}td_{p}(G)$ topologically.

Following [28], call a topological Abelian group $G$ strongly $p$-dense if
there exists $k\in{\mathbb{N}}^{*}$ such that
$p^{k}td_{p}(\widetilde{G})\subseteq G$. This technical property turns out to
be the key towards complete understanding of minimality of powers:

######  Theorem 2.5.

[28, Theorem 3.5] For a minimal Abelian group $G$ the following are
equivalent:

  * (a)

all powers of $G$ are minimal;

  * (b)

$G^{\mathfrak{c}}$ is minimal;

  * (c)

$G$ is strongly $p$-dense for every prime $p$.

For the proof of Theorem A we need the following lemma:

######  Lemma 2.6.

If a sequentially complete Abelian group $G$ is strongly $p$-dense for some
prime $p$, then $G\supseteq c(\widetilde{G})$.

###### Proof.

In fact $c(\widetilde{G})$ is connected hence divisible, therefore
$td_{p}(c(\widetilde{G}))$ is divisible as well. Now strong $p$-density
implies $td_{p}(c(\widetilde{G}))\subseteq G$. Since
$td_{p}(c(\widetilde{G}))$ is sequentially dense in $c(\widetilde{G})$ by [13,
Theorem 2.9(a)], it follows also that $G\cap c(\widetilde{G})$ is sequentially
dense in $c(\widetilde{G})$. Now the sequential completeness of $G$ yields
$G\supseteq c(\widetilde{G})$. ∎

The next lemma resolves the problem of minimality for finite products of
sequentially complete minimal groups:

######  Lemma 2.7.

Let $G$ be a sequentially complete minimal group. Then $G\times H$ is minimal
for every minimal group $H$. In particular, any finite product of sequentially
complete minimal groups is minimal.

###### Proof.

Fix an arbitrary prime $p$ and pick $x\in td_{p}(G)$. It suffices to check
that $x$ is contained is a compact subgroup of $G$, then [8, Proposition
6.1.13] (see also [9, Proposition 4.1]) applies. The case when $C=\langle
x\rangle$ is finite is trivial. Assume that $C$ is infinite. Then $C$ is
metizable, being isomorphic to ${\mathbb{Z}}$ equipped with the $p$-adic
topology. Hence its compact closure $\overline{C}$ in $\widetilde{G}$ is
contained in $G$, since $G$ is sequentially complete. ∎

The minimality is not preserved by taking quotients (even quotients of minimal
group with respect to finite subgroups may fail to be minimal). In item (b) of
the next lemma one can find an instance when minimality is preserved by taking
quotients. In item (a) we recall a “three space property” from [18].

######  Proposition 2.8.

Let $G$ be a topological Abelian group and let $K$ be a compact subgroup of
$G$. Then

  * (a)

[18] if $G/K$ is minimal, then $G$ is minimal;

  * (b)

if $G$ is minimal and $K$ is connected, then $G/K$ is minimal.

###### Proof.

Let us only mention that (a) was proved in [18] without the assumption that
$G$ is Abelian and compactness of $K$ is relaxed to the conjunction of
minimality and completeness.

(b) First we note that $G/K$ is precompact, by Theorem 2.1. Since $K$ is
compact, this implies that $G$ is precompact too, so $\widetilde{G}$ is
compact and $\widetilde{G}/K\cong\widetilde{G/K}$. In order to apply Lemma 2.3
and Theorem 2.1 we need to check first that $Soc(\widetilde{G/K})\leq G/K$.
Pick $a^{\prime}=a+K\in Soc(\widetilde{G/K})$. Then $pa\in K$. Since $K$ is a
connected compact Abelian group, $K$ is divisible. Hence, $pa=pb$, for some
$b\in K$. Then $t=a-b\in Soc(\widetilde{G})\leq G$, by Lemma 2.3 and Theorem
2.1, since $G$ is minimal. Therefore, $a^{\prime}=t+b+K=t+K\in G/K$. In order
to conclude with the application of Lemma 2.3, consider a subgroup
${\mathbb{Z}}_{p}\cong{\mathbb{N}}^{*}\leq\widetilde{G}/K$ and put
$q(N_{1})=N$, where $q:\widetilde{G}\to\widetilde{G}/K$ is the canonical
homomorphism. In order to see that the short exact sequence $0\to K\to L\to
N\to 0$ splits, consider its dual short exact sequence
$0\to\widehat{N}\to\widehat{N}_{1}\to\widehat{K}\to 0$ which splits, due to
the divisibility of $\widehat{N}\cong{\mathbb{Z}}(p^{\infty})$. Hence, $L\cong
K\times N_{1}$, where $N_{1}\cong{\mathbb{Z}}_{p}$ is a closed subgroup of $L$
with $q(N_{1})=N$, so clearly the restriction $q\restriction_{N_{1}}:N_{1}\to
N$ is an isomorphism. Again by Lemma 2.3 and Theorem 2.1, applied to the
minimal group $G$, $N_{1}\cap G\neq\\{0\\}$. Since, $q\restriction_{N_{1}}$ is
an isomorphism, this implies $N\cap(G/K)\neq\\{0\\}$. ∎

######  Corollary 2.9.

Let $G$ be a topological Abelian group such that $c(G)$ is compact. Then $G$
is minimal if and only if $G/c(G)$ is minimal.

Now we prove that the property (3) from Fact 2.4 can be extended to
sequentially complete minimal groups:

######  Theorem 2.10.

Let $G$ be a sequentially complete minimal totally disconnected Abelian group.
Then for every $p\in{\mathbb{P}}$ the subgroup $td_{p}(G)$ of $G$ is closed
and $G\cong\prod_{p\in{\mathbb{P}}}td_{p}(G)$ topologically.

###### Proof.

The completion $\widetilde{G}$ of $G$ is compact by Prodanov-Stoyanov’s
Theorem 2.1. Moreover, $\widetilde{G}$ is totally disconnected by Theorem C
(its proof does not depend on the proof of the current theorem). Hence,
$\widetilde{G}=\prod_{p}K_{p}$, where $K_{p}=td_{p}(\widetilde{G})$ for every
$p\in{\mathbb{P}}$, by Fact 2.4(3). Hence, for each $p\in{\mathbb{P}}$ the
subgroup $td_{p}(G)=G\cap K_{p}$ of $G$ is closed, by items (1b) and (3) of
Fact 2.4. Since $G$ is sequentially complete, to prove that
$P:=\prod_{p\in{\mathbb{P}}}td_{p}(G)\leq G$ it suffices to check that the
subgroup $S=\bigoplus_{p\in{\mathbb{P}}}td_{p}(G)$ of $G$ is sequentially
dense in $P$. In fact, let $x=(x_{p})\in P$. Then obviously
$x\in\prod_{p\in{\mathbb{P}}}\langle x_{p}\rangle$. As this subgroup of $P$ is
metrizable, $x\in\overline{\bigoplus_{p\in{\mathbb{P}}}\langle x_{p}\rangle}$
and $\bigoplus_{p\in{\mathbb{P}}}\langle x_{p}\rangle\subseteq S$, this proves
the sequential density of $S$ in $P$.

In order to prove the desired inclusion $G\leq P$ it suffices to see that the
canonical projection $\pi_{p}:\prod_{q\in{\mathbb{P}}}K_{q}\to K_{p}$
satisfies $\pi_{p}(G)\leq td_{p}(G)$, where we consider $\pi_{p}(G)$, as well
as $K_{p}=\pi_{p}(K)$, as subgroups of the product $K$. Since,
$td_{p}(G)=G\cap K_{p}$ and obviously $\pi_{p}(G)\leq K_{p}$, it suffices to
check that $\pi_{p}(G)\leq G$,

To prove this assertion take $x=(x_{q})_{q\in{\mathbb{P}}}\in G$, fix a prime
$p$ and let $p_{1},\ldots,p_{n},\ldots$ be an enumeration of the set
${\mathbb{P}}\setminus\\{p\\}$. Then we can consider $y=\pi_{p}(x)$ as the
element of the product, such that $\pi_{p}(y)=x_{p}$, while $\pi_{p_{n}}(y)=0$
for each $n\in{\mathbb{N}}^{*}$. We have to prove that $y\in G$. By the
Chinese theorem of remainders that there exists a sequence $\\{k_{n}\\}$ in
${{\mathbb{Z}}}$ such that

$k_{n}\equiv 1\ (\hbox{mod }p^{n})\ \ \hbox{ and }\ \ k_{n}\equiv 0\
(\hbox{mod }(p_{1}\ldots p_{n})^{n})$

for each $n\in{\mathbb{N}}^{*}$. Then $k_{n}\rightarrow 1$ in the $p$-adic
topology of ${\mathbb{Z}}$ and $k_{n}\rightarrow 0$ in the $p_{m}$-adic
topology of ${\mathbb{Z}}$ for each $m\in{\mathbb{N}}^{*}$. This yields

(2.1) $\lim_{n}k_{n}x_{p}=x_{p}\ \ \mbox{ in }\ \ A_{p}\ \ \mbox{ and }\ \
\lim_{n}k_{n}x_{p_{m}}=0\ \ \mbox{ in }\ \ A_{p_{m}}\ \ \mbox{ for each }\ \
m\in{\mathbb{N}}^{*}$

since the topology of $A_{q}$ is coarser than the $q$-adic topology of $A_{q}$
for every $q\in{\mathbb{P}}$. Since $G$ is sequentially complete,
$\lim{k_{n}}x\in G$. On the other hand, (2.1) entails $y=\lim{k_{n}}x\in G$. ∎

We denote by ${\mathbb{K}}$ the (compact) Pontryagin dual of the discrete
group ${\mathbb{Q}}$, i.e., ${\mathbb{K}}$ is the group of all homomorphisms
${\mathbb{Q}}\to{\mathbb{T}}$ equipped with the topology of pointwise
convergence. Since the Pontryagin dual of a compact torsion-free Abelian group
is a discrete divisible Abelian group, every compact torsion-free Abelian
group is a product of copies of ${\mathbb{K}}$ and the groups
${\mathbb{Z}}_{p}$. Hence, the next proposition reduces various problems
related to minimal Abelian groups $G$ to the case of dense minimal subgroups
of the powers ${\mathbb{K}}^{\alpha}$ or ${\mathbb{Z}}_{p}^{\alpha}$, where
$\alpha=w(G)$.

######  Proposition 2.11.

Let $G$ be a minimal Abelian group. Then there exists a compact torsion-free
Abelian group $K$ with $w(K)=w(G)$, a dense minimal subgroup $G_{1}$ of $K$
and a compact totally disconnected subgroup $N$ of $G_{1}$ such that:

  * (a)

the quotient group $G_{1}/N$ is isomorphic to $G$ and consequently
$K/N\cong\widetilde{G}$;

  * (b)

for every $p\in{\mathbb{P}}$ $G$ is strongly $p$-dense if and only if $G_{1}$
is strongly $p$-dense;

  * (c)

$G_{1}$ is countably compact (resp., sequentially complete, $\omega$-bounded,
totally disconnected) if $G$ is countably compact (resp., sequentially
complete, $\omega$-bounded, totally disconnected).

In case $G$ is a minimal quasi $p$-torsion group for some prime $p$, then
$K={\mathbb{Z}}_{p}^{w(G)}$.

###### Proof.

(a) and the part of (c) about sequential completeness are proved in [13,
Proposition 2.7]. The remaining part of (c) (concerning total
disconnectedness, countable compactness and $\omega$-boundedness) is clear.

To check (b) denote by $q:K\to\widetilde{G}$ the quotient map. Since
$q(td_{p}(K)=td_{p}(\widetilde{G})$ ([8]), strong $p$-density of $G_{1}$
provides $k\in{\mathbb{N}}$ with $p^{k}td_{p}(K)\leq G_{1}$, which obviously
implies strong $p$-density of $G$. If $G$ is strongly $p$-dense, say
$p^{k}td_{p}(\widetilde{G})\leq G$, then for $x\in td_{p}(G_{1})$ one has
$q(x)\in td_{p}(\widetilde{G})$, so $p^{k}q(x)=q(p^{k}x)\in G$. Therefore,
$p^{k}x\in G_{1}$. ∎

######  Corollary 2.12.

If $G$ is a sequentially complete minimal Abelian group with
$c(G)=c(\widetilde{G})$, then $G/c(G)$ is sequentially complete and minimal.

###### Proof.

By Proposition 2.11 there exists a compact torsion-free Abelian group $K$ with
$w(K)=w(G)$, a dense sequentially complete minimal subgroup $G_{1}$ of $K$ and
a compact totally disconnected subgroup $N$ of $G_{1}$ such that $G_{1}/N\cong
G$. Let $f:K\to K/N\cong\widetilde{G}$ be the canonical projection. Since $f$
is surjective, we have $f(c(K))=c(\widetilde{G})$. Since
$c(G)=c(\widetilde{G})$, $G_{1}$ contains $c(K)$. Moreover, as $K$ is torsion-
free, one has $K=c(K)\times D$, where $D$ is a compact totally disconnected
subgroup of $K$. Then $G_{1}=c(K)\times G_{2}$, where $G_{2}=D\cap G_{1}$ is a
closed subgroup of $G_{1}$. Hence $G_{2}$ is sequentially complete and
minimal. As $G/c(G)\cong G_{1}/c(K)=(c(K)\times G_{2})/c(K)\cong G_{2}$, we
deduce that $G/c(G)$ is sequentially complete and minimal. ∎

Let us note that as far as only minimality of $G/c(G)$ is concerned, Corollary
2.9 applies, so that only compactness of $c(G)$ suffices to this end (so the
equality $c(G)=c(\widetilde{G})$ is not needed).

###  2.2. Proof of Theorems A, B, C, D and Corollary 1.3

The major step towards the proof of Theorem A concerns the totally
disconnected case. The following proposition and its immediate corollary go in
this direction:

######  Proposition 2.13.

A totally disconnected sequentially complete minimal Abelian group is strongly
$p$-dense for every prime $p$.

Applying Theorem 2.5, one obtains:

######  Corollary 2.14.

All powers of a totally disconnected sequentially complete minimal Abelian
group are minimal.

We split the proof of Proposition 2.13 in several steps.

The next lemma is a local version of Proposition 2.13 and provides an
essential tool for the proof of Theorem C(a) as well. In the case of subgroups
of ${{\mathbb{Z}}}_{p}^{\alpha}$ (which is the key to the proof in the general
case of the lemma), it was proved in [15, Main Lemma] for $\omega$-bounded
groups $G$ by means of a specific functorial correspondence, based on
Pontryagin duality, between precompact groups covered by their compact
subgroups and linearly topologized groups introduced in [30]. This duality
technique essentially used the assumption of $\omega$-boundedness and cannot
work even under the assumption of countable compactness. The proof given here
in the sequentially complete case is completely different, it has purely
topological nature.

######  Lemma 2.15.

Let $p$ be a prime number. If $G$ is a dense sequentially complete minimal
subgroup $G$ of a pro-$p$-group $A$, then there exists $k\in{\mathbb{N}}$ such
that $p^{k}A\subseteq G$.

###### Proof.

We first show that the proof can be reduced to the case when
$A={{\mathbb{Z}}}^{\alpha}_{p}$. By Proposition 2.11, $A$ is a quotient of
$B={{\mathbb{Z}}}^{\alpha}_{p}$, with $\alpha=w(A)$ and $G_{1}=f^{-1}(G)$ is a
sequentially complete minimal subgroup of $B$. If there exists
$k\in{\mathbb{N}}^{*}$ such that $p^{k}B\subseteq G_{1}$, then clearly,
$p^{k}A\subseteq G$.

Obviously, $A={{\mathbb{Z}}}^{\alpha}_{p}$ is a ${\mathbb{Z}}_{p}$-module.
Moreover, since for $a\in G$ the cyclic ${\mathbb{Z}}_{p}$-submodule
${\mathbb{Z}}_{p}a\cong{\mathbb{Z}}_{p}$ is compact metrizable, so it is
contained in the sequentially complete subgroup $G$. Hence, $G$ is a submodule
of $A$. By the minimality of $G$ and Theorem 2.2, for $0\neq x\in A$ the
cyclic ${\mathbb{Z}}_{p}$-submodule ${\mathbb{Z}}_{p}x=\overline{\langle
x\rangle}$ non-trivially meets $G$, hence $A/G$ is torsion
${\mathbb{Z}}_{p}$-module, so a $p$-group. We need to prove that it is a
bounded $p$-group. Clearly, it is enough to prove that for every sequence
$\\{x_{k}:k\in{\mathbb{N}}^{*}\\}\subseteq A$ there exists
$n\in{\mathbb{N}}^{*}$ (independent on $k$) such that $p^{n}x_{k}\in G$ for
all $k$.

Consider the group ${\mathbb{Z}}_{p}^{\omega}$ equipped with the $p$-adic
topology $\tau_{p}$. A base of the neighbourhoods of 0 in the topology
$\tau_{p}$ is given by is given by the open subgroups
$V_{n}=p^{n}{\mathbb{Z}}_{p}^{\omega}$, $n\in{\mathbb{N}}^{*}$. Then
${\mathbb{Z}}_{p}^{\omega}$ is a complete metrizable topological group. Let
$E\subseteq{\mathbb{Z}}_{p}^{\omega}$ be the set of all sequences
$\\{a_{k}:k\in{\mathbb{N}}^{*}\\}$ with $\lim a_{k}=0$.

######  Claim 2.16.

$E$ is a closed pure submodule of ${\mathbb{Z}}_{p}^{\omega}$, i.e., $E\cap
p^{m}{\mathbb{Z}}_{p}^{\omega}=p^{m}E$ for all $m\in{\mathbb{N}}^{*}$.

###### Proof.

Obviously, $E$ is a submodule of ${\mathbb{Z}}_{p}^{\omega}$. To check that
$E$ is closed, pick $c=(c_{k})\in{\mathbb{Z}}_{p}^{\omega}\setminus E$. Then
$c_{k}\not\to 0$, so there exists $m$ such that for every
$k\in{\mathbb{N}}^{*}$ there exists $l>k$ such that $c_{l}\not\in
p^{m}{\mathbb{Z}}_{p}$. Then $(c+V_{m})\cap E=\emptyset$. Indeed, if
$a=(a_{k})\in E$, then there exists $t>k$ such that $a_{s}\in
p^{m}{\mathbb{Z}}_{p}$ for all $s\geq t$. Taking now $s\geq t$ such that
$c_{s}\not\in V_{m}$ we have $a_{s}-c_{s}\not\in p^{m}{\mathbb{Z}}_{p}$, as
$a_{s}\in p^{m}{\mathbb{Z}}_{p}$. Hence $a-c\not\in V_{m}$ and $a\not\in
c+V_{m}$. This proves that $(c+V_{m})\cap E=\emptyset$.

It remains to check that $E\cap p^{m}{\mathbb{Z}}_{p}^{\omega}=p^{m}E$ for all
$m\in{\mathbb{N}}^{*}$. To check the non-trivial inclusion $E\cap
p^{m}{\mathbb{Z}}_{p}^{\omega}\leq p^{m}E$ for $m\in{\mathbb{N}}^{*}$ pick
$a=(a_{k})\in E\cap p^{m}{\mathbb{Z}}_{p}^{\omega}$. Then $a_{k}=p^{m}b_{k}$
for some $b_{k}\in{\mathbb{Z}}_{p}$ and for all $k$. Since obviously $b_{k}\to
0$ in ${\mathbb{Z}}_{p}$, $b=(b_{k})\in E$, so $a\in p^{m}b\in p^{m}E$. ∎

Now we define a map $\varphi:E\to A$ putting $\varphi(e_{n})=x_{n}$, where
$\\{e_{n}:n\in{\mathbb{N}}^{*}\\}$ is the canonical base of the dense
submodule $E^{\prime}={\mathbb{Z}}_{p}^{(\omega)}$ of $E$. This assignment can
be uniquely extended to a ${\mathbb{Z}}_{p}$-homomorphism
$\varphi:E^{\prime}\to A$ which is obviously continuous, so can be further
(uniquely) extended to a continuous ${\mathbb{Z}}_{p}$-homomorphism
$\varphi:E\to A$ satisfying $\\{a_{k}\\}\buildrel{\varphi}\over{\mapsto}\sum
a_{k}x_{k}$.

The ${\mathbb{Z}}_{p}$-submodule $F=\varphi^{-1}(G)$ of $E$ is sequentially
closed (and hence closed) and $E/F$, being isomorphic to a submodule of $A/G$,
is a $p$-group. Therefore, for every $s\in E$ there exists
$n\in{\mathbb{N}}^{*}$ such that $p^{n}s\in F$, so
$E/F=\bigcup_{n\in{\mathbb{N}}^{*}}(E/F)[p^{n}]$ and $(E/F)[p^{n}]$ is closed
in $E/F$ equipped with the quotient topology. Let $q:E\to E/F$ be the
canonical homomorphism. Then $E_{n}=q^{-1}((E/F)[p^{n}])$ is closed in $E$ and
$E=\bigcup_{n\in{\mathbb{N}}^{*}}E_{n}$. By the Baire category theorem implies
$E_{k}$ has a non-empty interior for some $k\in{\mathbb{N}}^{*}$. Since
$E_{k}$ is a subgroup, this yields that $F_{k}$ is open. Hence there exists an
open subgroup $E\cap V_{m}=p^{m}E$ in $E$ such that
$p^{k}(p^{m}E)=p^{m+k}E\subseteq F$ for some $k\in{\mathbb{N}}^{*}$. This
means, that for every sequence $\\{a_{i}\\}\subseteq p^{m+k}E$ one has
$\varphi(\\{a_{i}\\})\in G$. In particular, for every $i\in{\mathbb{N}}^{*}$
we have $p^{m+k}x_{i}=\varphi(p^{m+k}e_{i})\in G$. ∎

Now we prove first Theorem C that will be used below in the proof of
Proposition 2.13 and Theorem B.

Proof of Theorem C. Both (a) and (b) are proved in [13, Theorem 3.6] in full
detail although the proof is quite long and heavy. Furthermore, (a) and (b)
are not articulated, with the blanket assumption that $w(c(\widetilde{G}))$ is
not Ulam measurable. We preferred to split (a) and (b) so that it becomes
clear where the relevant assumption that $w(c(\widetilde{G}))$ is not Ulam
measurable is used (only in item(b)), as well as the crucial role of Lemma
2.15 in the proof of (a).

(a) The proof of the fact that $c(G)=c(\widetilde{G})$ whenever $c(G)$ is
compact can be found in Steps 1 and 3 of the proof of [13, Theorem 3.6]. Both
steps essentially use Lemma 2.15, in the particular (but essential) case when
$A$ is a power of ${\mathbb{Z}}_{p}$.

We prefer to point out a self-contained argument for deducing the remaining
part of (a) from the equality $c(G)=c(\widetilde{G})$, rather than refer to
[13]. Indeed, Corollary 2.12 implies that this equality yields minimality and
sequential completeness of $G/c(G)$.

(b) The proof of this item is Step 2 in the proof of [13, Theorem 3.6].
$\Box$.

Proof of Proposition 2.13. We have to prove that a totally disconnected
sequentially complete minimal Abelian group $G$ is strongly $p$-dense for
every prime $p$.

By Theorems 2.1 and C, the completion $K$ of $G$ is compact and totally
disconnected, hence $K=\prod_{p}K_{p}$, where $K_{p}=td_{p}(K)$ is a
pro-$p$-group, according to Fact 2.4(3). By Theorem 2.10, $G=\prod_{p}G_{p}$,
where $G_{p}=td_{p}(G)=G\cap K_{p}$ is a dense essential subgroup of $K_{p}$.
Moreover, $G_{p}$ is sequentially complete, being a closed subgroup of $G$. By
Lemma 2.15, there exists $k_{p}\in{\mathbb{N}}^{*}$ with
$p^{k_{p}}K_{p}\subseteq G_{p}\subseteq G$. $\Box$

Proof of Theorem A. We have to prove that for a minimal sequentially complete
Abelian group $G$ the following are equivalent:

  * (a)

all powers of $G$ are minimal;

  * (b)

$G^{\omega}$ is minimal.

  * (c)

$G$ contains $c(\widetilde{G})$; hence, $c(G)=c(\widetilde{G})$ is compact.

To prove the implication (b) $\to$ (a) assume that $G^{\omega}$ is minimal.
Since for every prime $p$ every closed monothetic subgroup of $td_{p}(G)$ is
compact, it follows from [5] (see also [8, Corollary 6.2.8]) that all powers
of $G$ are minimal, i.e., (a) holds.

To prove the the implication (a) $\to$ (c) assume that all powers $G^{\alpha}$
are minimal. Then by Theorem 2.5 $G$ is strongly $p$-dense for some prime $p$.
By Lemma 2.6, $G$ contains $K=c(\widetilde{G})$. Since $K$ is divisible, this
yields $td_{p}(K)$ is divisible too. Hence $G$ contains $td_{p}(K)$. According
to [13, Theorem 2.9(a)] $td_{p}(K)$ is sequentially dense in $K$. Therefore,
the sequential completeness of $G$ implies $K\subseteq G$. In particular,
$c(G)=K$ is compact.

For the proof of the implication (c) $\to$ (b) assume that
$c(G)=c(\widetilde{G})$ holds true. By Corollary 2.12, $G/c(G)$ is minimal and
sequentially complete. Since $G/c(G)$ is totally disconnected, Corollary 2.14
yields that $(G/c(G))^{\omega}$ is minimal. Since $(G/c(G))^{\omega}\cong
G^{\omega}/(c(G))^{\omega}$ and $c(G)^{\omega}$ is compact, we deduce that
$G^{\omega}$ is minimal, by Proposition 2.8. $\Box$

Proof of Corollary 1.3. We have to prove that if $\\{G_{i}\\}_{i\in I}$ is a
family of sequentially complete Abelian groups, then $G=\prod_{i\in I}G_{i}$
is minimal if and only if every countable subproduct is minimal and in such a
case there exists a co-finite subset $J\subseteq I$ such that all groups
$G_{i}^{\omega}$, $i\in J$, are minimal.

The first part follows from [8, Theorem 6.2.7]. According to the same theorem,
there exists a co-finite subset $J\subseteq I$ such that all groups $G_{i}$,
$i\in J$, are strongly $p$-dense. According to Theorem 2.5 this yields
minimality of $G_{i}^{\omega}$. $\Box$

Proof of Theorem B. The implication (c) $\to$ (b) is trivial. The implication
(b) $\to$ (a) follows from Theorem C.

(a) $\to$ (c) Let $\alpha$ be Ulam measurable and let $I$ be a set of
cardinality $\alpha$. Since the cardinal $\alpha$ is Ulam measurable, there
exists a countably complete ultrafilter ${\mathcal{F}}$ on $I$. Following [8,
Proposition 3.6.1], in the sequel we use the following properties of
${\mathbb{K}}$ and related notation. Taking the dual of the short exact
sequence $0\to{\mathbb{Z}}\to{\mathbb{Q}}\to{\mathbb{Q}}/{\mathbb{Z}}\to 0$
one obtains a short exact sequence
$0\to\widehat{{\mathbb{Q}}/{\mathbb{Z}}}\to{\mathbb{K}}=\widehat{{\mathbb{Q}}}\to\widehat{{\mathbb{Z}}}={\mathbb{T}}\to
0$. Here we denote by ${\mathbb{H}}$ the closed subgroup
$\widehat{{\mathbb{Q}}/{\mathbb{Z}}}\cong\prod_{p}{\mathbb{Z}}_{p}$ so
${\mathbb{K}}$, so that
${\mathbb{H}}_{p}:=td_{p}({\mathbb{H}})\cong{\mathbb{Z}}_{p}$, up to
isomorphism.

The subgroup
$B_{\mathcal{F}}=\bigcup_{S\in{\mathcal{F}}}{\mathbb{K}}^{I\setminus S}$ of
${\mathbb{K}}^{I}$ is dense, since ${\mathbb{K}}^{(I)}\leq B_{\mathcal{F}}$
and the former subgroup is dense. Since the filter ${\mathcal{F}}$ is
countably complete, the family of subgroup $\\{{\mathbb{K}}^{I\setminus
S}\\}_{S\in{\mathcal{F}}}$ is $\omega$-directed, hence the subgroup
$B_{\mathcal{F}}$ is $\omega$-bounded. Then the group
$G_{\mathcal{F}}:={\mathbb{H}}^{I}+B_{\mathcal{F}}$ is $\omega$-bounded as
well, as ${\mathbb{H}}^{I}$ is compact. So far we proved that
$G_{\mathcal{F}}$ is a dense $\omega$-bounded subgroup of ${\mathbb{K}}^{I}$.
To see that it is not compact, it suffices to note that it is a proper
subgroup of ${\mathbb{K}}^{I}$ (indeed, if
$a\in{\mathbb{K}}\setminus{\mathbb{H}}$, then the point ${\bf a}$ of
${\mathbb{K}}^{I}$ with all coordinates equal to $a$ does not belong to
$G_{\mathcal{F}}$).

It only remains to check that $G_{\mathcal{F}}$ is minimal. According to
Theorem 2.2, we have to prove that $G_{\mathcal{F}}$ is essential in
$\widetilde{G}_{\mathcal{F}}={\mathbb{K}}^{I}$. By Lemma 2.3, it suffices to
check that for every prime $p$ and for every $x=(x_{i})\in
td_{p}({\mathbb{K}}^{I})=(td_{p}({\mathbb{K}}))^{I}$ there exists
$n\in{\mathbb{N}}^{*}$ such that $p^{n}x\in G_{\mathcal{F}}$. Let
$supp(x)=\\{i\in I:x_{i}\neq 0\\}$. For $n\in{\mathbb{N}}^{*}$ set
$T_{n}=\\{i\in supp(x):p^{n}x_{i}\not\in{\mathbb{H}}_{p}\\}$. Obviously
$T_{1}\supseteq T_{2}\supseteq\ldots$ is a chain in $X$ and $\bigcap
T_{n}=\emptyset$ as $i\in supp(x)$ yields there exists $k$, so that $i\not\in
T_{k}$, as $x_{i}\in
td_{p}({\mathbb{K}})=\bigcup_{k=0}^{\infty}p^{-k}{\mathbb{H}}_{p}$. Hence

(2.2) $I=\bigcup_{n=1}^{\infty}I\setminus T_{n}.$

Since ${\mathcal{F}}$ is a countably complete ultrafilter, it follows from
(2.2) that $I\setminus T_{n_{0}}\in{\mathcal{F}}$ for some
$n_{0}\in{\mathbb{N}}^{*}$. Hence ${\mathbb{K}}^{T_{n_{0}}}\subseteq
B_{\mathcal{F}}\subseteq G_{\mathcal{F}}$. Split
$x=x^{\prime}+x^{\prime\prime}$, such that $supp(x^{\prime})\cap
supp(x^{\prime\prime})=\emptyset$ and $supp(x^{\prime})\subseteq T_{n_{0}}$.
Then $supp(x^{\prime\prime})\subseteq I\setminus T_{n_{0}}$ so that
$p^{n_{0}}x^{\prime\prime}\in{{\mathbb{H}}^{I}}\subseteq G_{\mathcal{F}}$,
while $x^{\prime}\in{\mathbb{K}}^{T_{n_{0}}}\subseteq G_{\mathcal{F}}$.
Consequently, $p^{n_{0}}x\in G_{\mathcal{F}}$.

To prove the last assertion of the theorem assume now that $\alpha$ is
measurable. Then we can choose the filter ${\mathcal{F}}$ complete, i.e.,
stable under intersections of families of cardinality $<\alpha$. As above, the
family of subgroup $\\{{\mathbb{K}}^{I\setminus S}\\}_{S\in{\mathcal{F}}}$ is
$\beta$-directed for every $\beta<\alpha$, hence the subgroup
$B_{\mathcal{F}}=\bigcup_{S\in{\mathcal{F}}}{\mathbb{K}}^{I\setminus S}$ of
${\mathbb{K}}^{I}$, defined as above, is $\beta$-bounded for every
$\beta<\alpha$. Then also the group
$G_{\mathcal{F}}={\mathbb{H}}^{I}+B_{\mathcal{F}}$ is $\beta$-bounded for
every $\beta<\alpha$. The rest of the proof works in the same way. $\Box$

The following theorem provides a proof of a stronger version of Theorem D:

######  Theorem 2.17.

Let $G$ be a sequentially complete Abelian such that $G^{\omega}$ is minimal.
Then then for every $p\in{\mathbb{P}}$ there exists a
$k_{p}\in{\mathbb{N}}^{*}$ such that $G$ contains the compact subgroup
$N=\overline{\langle p^{k_{p}}td_{p}(\widetilde{G}):p\in{\mathbb{P}}\rangle}$
(the closure taken in $\widetilde{G}$) and
$G/N\cong\prod_{p\in{\mathbb{P}}}B_{p}$, where $B_{p}$ is a bounded
$p$-torsion minimal group for every prime $p$.

###### Proof.

By Theorem A, $c(G)=c(\widetilde{G})$ is compact. By Corollary 2.12,
$G_{1}:=G/c(G)$ is minimal and sequentially complete. Put
$A=\widetilde{G}/c(G)$ and let $q:\widetilde{G}\to\widetilde{G}/c(G)$ be the
canonical homomorphism. Clearly $A$ is the completion of $G_{1}$. Put for
brevity $A_{p}=td_{p}(A)$. By Fact 2.4(3), $A=\prod_{p}A_{p}$, since $A$ is
totally disconnected. Proposition 2.13 implies that for every prime $p$ there
exists $k_{p}\in{\mathbb{N}}^{*}$ such that

(2.3) $N_{p}:=p^{k_{p}}A_{p}\leq G_{p}$

Hence, $S_{1}=\bigoplus_{p\in{\mathbb{P}}}pN_{p}\leq G_{1}$. Since $S_{1}$ is
sequentially dense in the closed subgroup
$N_{1}=\prod_{p\in{\mathbb{P}}}pN_{p}$ of $A$ and $G_{1}$ is sequentially
complete, we deduce that $N_{1}\leq G_{1}$. Hence, $N=q^{-1}(N_{1})\leq G$. By
Fact 2.4, $q(td_{p}(\widetilde{G}))=A_{p}$ for every prime $p$, hence the
subgroup $S=\langle p^{k_{p}+1}td_{p}(\widetilde{G}):p\in{\mathbb{P}}\rangle$
of $\widetilde{G}$ satisfies $q(S)=S_{1}\leq N_{1}$, so $S\leq N\leq G$.
Hence, $S\leq L:=\overline{S}\leq N$ and $S_{1}=q(S)\leq q(L)\leq
q(N)=N_{1}=\overline{S_{1}}$. Since $q(L)$ is compact, we deduce that
$q(L)=q(N)$. As $\ker q=c(\widetilde{G})\leq L\leq N$ by (the proof of) Lemma
2.6, we conclude that $L=N$. This proves the first assertion of the theorem.

To prove the second one we put $B_{p}=A_{p}/pN_{p}=A_{p}/p^{k_{p}+1}A_{p}$ and
note that $A/N_{1}\cong\prod_{p\in{\mathbb{P}}}B_{p}$. Since $G/N\cong
G_{1}/N_{1}$, it is enough to prove that $G_{1}/N_{1}$ is minimal. According
to Theorem 2.2, this amounts to check that $G_{1}/N_{1}$ is a an essential
subgroup of $A/N_{1}$. Since for every prime $p$ the group $B_{p}$ is a
bounded $p$-group, it suffices to check that $Soc(B_{p})\leq G_{1}/N_{1}$,
according to Lemma 2.3.

The minimality of $G_{1}$ and Lemma 2.3 yield that
$Soc(A)=\bigoplus_{p\in{\mathbb{P}}}A[p]\leq G_{1}$. By virtue of (2.3), we
obtain

(2.4) $A[p]+p^{k_{p}}A_{p}\leq G_{1}.$

To check that $Soc(B_{p})\leq G_{1}/N_{1}$ pick $\overline{a}=a+pN_{p}\in
Soc(B_{p})$, then $p\overline{a}=0$ in $B_{p}$, so $pa\in pN_{p}$.
Consequently, $pa=p^{k_{p}+1}b$ for some $b\in A_{p}$. Hence,
$t:=a-p^{k_{p}}b\in A[p]$. Now (2.4) gives $a=t+p^{k_{p}}b\in G$. Therefore,
$\overline{a}\in G_{1}/N_{1}$. ∎

##  3\. Final comments and questions

The groups considered in this section are not necessarily Abelian.

Sequential completeness is not preserved under taking continuous homomorphic
images. Following [14], call a group $G$ $h$-sequentially complete if all
continuous homomorphic images of $G$ are sequentially complete. Countably
compact groups are obviously $h$-sequentially complete. Nilpotent sequentially
$h$-complete groups are precompact [14, Theorem 3.6], while any precompact
sequentially $h$-complete group is pseudocompact [14, Theorem 3.9].

######  Question 3.1.

  * (1)

Find an example of a precompact $h$-sequentially complete group that is not
countably compact.

  * (2)

Find an example of a pseudocompact sequentially complete group that is not
$h$-sequentially complete.

  * (3)

Find an example of a (precompact) sequentially complete group that has non-
sequentially-complete quotients.

  * (4)

Are precompact sequentially complete minimal groups pseudocompact?

  * (5)

Are sequentially complete minimal totally (hereditarily) disconnected groups
zero-dimensional?

In some cases (4) has a positive answer – when the group has large
hereditarily pseudocompact (in particular, countably compact) subgroups. In
view of [6, Theorem 1.2] (5) is true for $h$-sequentially complete groups. On
the other hand, it was shown in [13] that (5) is true for sequentially
complete minimal Abelian groups.

######  Question 3.2.

Which of the following properties of an infinite topological group guarantee
the existence of non-trivial converging sequences: i) countably compact and
minimal; ii) countably compact and totally minimal; iii) totally minimal?

The following question was left open in [13]:

######  Question 3.3.

Let $G$ be a sequentially complete minimal Abelian group. Is then $G/c(G)$
sequentially complete?

Finally, we recall an old open problem raised by the second named author
closely related to the topic of this paper. We now cancel our assumption that
all groups under consideration are Abelian.

######  Question 3.4.

(Uspenskij [31]) Is it true that arbitrary products of complete minimal groups
are minimal? What about infinite powers?

Note that the answer is positive in the Abelian case, since minimal complete
Abelian groups are compact, by Theorem 2.1. “Complete” in Question 3.4 means
complete with respect to the two-sided uniformity. The question also makes
sense for the case of Weil-complete groups, that is, groups complete with
respect to the left (or right) uniformity. A particular instance of such
groups are the locally compact ones. The above question seems to be open even
in the case of locally compact minimal groups (many instances of positive
answer in this case can be found in [26], yet no general result seems to be
available to the best of our knowledge).

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#  Incommensurate spin and orbital magnetism of Mn atomic chains on W(110)
surface

M. M. Bezerra-Neto1 Y. O. Kvashnin2 A. Bergman2 R. B. Muniz3 O. Eriksson2,4 M.
I. Katsnelson5 A. B. Klautau6 1Instituto de Engenharia e Geociências,
Universidade Federal do Oeste do Pará, Santarém, PA, Brazil  2Department of
Physics and Astronomy, Division of Materials Theory, Uppsala University, Box
516, SE-751 20, Sweden  3Instituto de Física, Universidade Federal Fluminense,
24210-346 Niterói, Rio de Janeiro, Brazil  4School of Science and Technology,
Örebro University, SE-701 82, Örebro, Sweden  5Institute for Molecules and
Materials, Radboud University, Heijendaalseweg 135, 6525AJ, Nijmegen,
Netherlands  6Faculdade de Física, Universidade Federal do Pará, Belém, PA,
Brazil

###### Abstract

Stabilization of unusual spin-orbit driven magnetic orderings are achieved for
chains of Mn atoms deposited on a W(110) substrate. First-principles
electronic structure calculations show that the ground state spin
configuration is non-collinear, forming spirals as a result of a competition
between nearest and next-nearest neighbour exchange interactions. The orbital
magnetic moments are also found to exhibit non-collinear ordering, that
interestingly for some systems is incommensurate with the spin arrangement. We
attribute such an exotic behaviour to the competition between the and spin-
orbital interaction and crystal-field splitting effects. Model calculations
based on this assumption reproduce the main findings observed in the first-
principles calculations.

##  I Introduction

Chiral spin textures, such as skyrmions Fert  _et al._ (2013); Romming  _et
al._ (2013) and spin spirals Yoshida  _et al._ (2012); Bode  _et al._ (2007);
Menzel  _et al._ (2012); Steinbrecher  _et al._ (2018), have attracted intense
interest not only because of the fundamental physics governing their origins
but also due to their potential applications in novel information storage
technologies. Ultrathin magnetic films deposited on heavy metal substrates are
good candidates for displaying complex spin textures. In general, the chiral
spin states observed in these systems are induced by an interplay between the
Dzyaloshinskii-Moriya (DM) interaction Dzyaloshinsky (1958); Moriya (1960),
the bi-linear Heisenberg exchange interaction, and the magnetocrystalline
anisotropy. Ultrathin layers of Mn/W(110) Yoshida  _et al._ (2012); Bode  _et
al._ (2007) are one of these cases, where a combination of the relatively
strong spin-orbit interaction of tungsten and the broken inversion symmetry at
the interfaces lead to a significant interface-induced DM interaction. As a
result, the ground-state spin magnetic moments display different types of spin
spirals, depending on the number of Mn atomic planes. For a single Mn
overlayer on W(110) the spin magnetic moments exhibit a cycloidal magnetic
order as observed in Ref. Bode  _et al._ , 2007. The Mn bilayer show a conical
spin-spiral state along the [001] direction Yoshida  _et al._ (2012) and in
both cases a unique rotational sense is enforced by the chiral DM interaction.

With the advent of spin-orbitronics, which seeks to explore phenomena
intrinsically related to the spin-orbit coupling, the development of model
Hamiltonians has gained momentum in recent years, and some new approaches have
been discussed in the literature Secchi  _et al._ (2015); Mankovsky  _et al._
(2020); Lászlóffy  _et al._ (2019); Grytsiuk  _et al._ (2020); Cardias  _et
al._ (2020a). In this scenario, orbital moments became a focus of studies and
have been explored in various aspects, such as: in the so-called orbital Hall
effect, where an external electric field induces an orbital angular momentum
current that flows in the transverse direction of the applied field Tanaka
_et al._ (2008); Kontani  _et al._ (2009); Go  _et al._ (2018), in magnons
that besides their spin magnetic moment may also carry orbital magnetic moment
Neumann  _et al._ (2020), as a means to exert torque in magnetic units Go and
Lee (2020), in the formation of Rashba-type surface band splittings that may
lead to chiral orbital angular momentum states Park  _et al._ (2011, 2013), as
well as in the connection between the anisotropy of the orbital magnetic
moment and the DM interaction Kim  _et al._ (2018), among others. A model
Hamiltonian based on realistic electronic structure that takes into account in
a general way all relevant interactions remains a considerable challenge. One
of the promising approaches is based on the consideration of small rotations
near a given magnetic configuration (“magnetic force theorem”) as was
previously suggested for purely spin rotations Liechtenstein  _et al._ (1987);
Katsnelson and Lichtenstein (2000). Its generalization for the systems with
unquenched orbital moments Secchi  _et al._ (2015); Pourovskii (2016) allows
to write explicit formulas for spin-spin, orbital-orbital, and spin-orbital
interactions but until now they were not applied to specific real materials.

Here, by means of first-principles calculations, we demonstrate the occurrence
of atomic scale spin spirals in Mn chains placed on top of a W(110) surface.
The choice of these systems was motivated by the remarkable magnetic
properties observed for Mn overlayers on W(110) Yoshida  _et al._ (2012); Bode
_et al._ (2007), as well as by the possibility of fabricating atomic chains of
transition metal atoms on heavy metal substrates. We have examined the ground
state spin and orbital magnetic configurations of chains and nanoribbons
(stripes) of Mn deposited along two distinct crystallographic directions of a
W(110) surface. We found that the ground-state spin configuration of the Mn
nanostructure changes with the increase of its width. The mono-atomic Mn chain
display a spin spiral with helical nature, whereas Mn ribbons with three atoms
in width show a cycloidal spin spiral state. Such chiral magnetic orders are
also revealed for the orbital moments in some cases, and depending on the
length and width of the Mn nanostructures, the spin and orbital spiral
configurations may exhibit different periods. We discuss the obtained magnetic
chirality in terms of effective low energy Hamiltonians that contain spin-
spin, orbital-orbital as well as as spin-orbital exchange. The exotic
behaviour concerning the incommensurate orbital spiral configurations with
respect to the spin arrangement is microscopically attributed to a competition
between crystal field and spin-orbit coupling, which are comparable in this
material. Our model calculations, based on this assumption, reproduce the main
findings observed in the first-principles calculations.

##  II Computational details

The density functional theory based calculations were performed using the
first principles real-space linear muffin-tin orbital method within the atomic
sphere approximation (RS-LMTO-ASA) Frota-Pessoâ (1992); Klautau and Frota-
Pessôa (2005); Bergman  _et al._ (2006a, 2007); Bezerra-Neto  _et al._ (2013);
Klautau and Frota-Pessôa (2004); Frota-Pessôa  _et al._ (2002); Bergman  _et
al._ (2006b); Igarashi  _et al._ (2012); Cardias  _et al._ (2016), which has
been generalized to treat non-collinear magnetism Bergman  _et al._ (2006b);
Cardias  _et al._ (2020b, a). This method is based on the LMTO-ASA formalism
Andersen (1975) and employs the recursion method Haydock (1980) to solve the
eigenvalue problem directly in real space. All calculations are fully self-
consistent and exchange-correlation effects were taken into account within the
local spin density approximation (LSDA) von Barth and Hedin (1972). We
included the spin-orbit coupling term $\lambda L\cdot S$ self-consistently at
each variational step, where $L$ and $S$ represent the orbital and spin
angular momentum operators, and $\lambda$ is the calculated spin-orbit
coupling strength Eriksson  _et al._ (1990); Frota-Pessôa (2004).

We have considered Mn nanostructures, with different geometries, supported on
a W(110) surface. The W(110) substrate is modeled by a cluster containing
approximately 8000 atoms positioned in a bcc lattice with the experimental
lattice parameter of W. In order to simulate the vacuum outside the bcc
surface and to treat charge transfers correctly, we included two overlayers of
empty spheres above the W surface. The calculations of the Mn adsorbed
nanochains have been performed by embedding the clusters as a perturbation on
the previously self-consistently converged W(110) surface. In order to achieve
an accurate description of the hybridization with the substrate, the Mn sites
and the closest shells of neighboring atoms (or empty spheres) around the Mn
nanostructures were recalculated self-consistently. The continued fraction in
the recursion method was appended with the Beer-Pettifor terminator Beer and
Pettifor (1984) after 30 recursion levels.

The Heisenberg bi-linear exchange interactions $J_{ij}$ and the DM vectors
$\vec{D}_{ij}$ were calculated with the use of the
Liechtenstein–Katsnelson–Antropov–Gubanov (LKAG) formula Liechtenstein  _et
al._ (1987); Katsnelson and Lichtenstein (2000), as implemented in the RS-
LMTO-ASA method Frota-Pessôa  _et al._ (2000); Szilva  _et al._ (2013, 2017);
Cardias  _et al._ (2020b, a). Both $J_{ij}$ and $D_{ij}$ have been obtained
from the ferromagnetic configuration of the nanoclusters. Their values are
then used to analyze the spin magnetic orderings explored in the full non-
collinear calculations.

##  III Results and Discussions

Fig. 1 illustrates the finite Mn nanostructures adsorbed on W(110) that were
investigated here. First, the ground-state spin magnetic configuration of
these nanostructures was investigated and, subsequently, focus was put on
their orbital magnetic orderings. We start the discussion with the mono-atomic
chain (hereafter identified as NW, for nanowire) comprising of 15 Mn atoms
deposited along the [$1\overline{1}1$] direction, having 3.8 nm in length.
Next, we focus on a double nanowire (DNW), constituted of two 9 atom long Mn
chains, also placed along the [$1\overline{1}1$] direction, with a length of
2.2 nm. Finally, two finite stripes with three Mn atoms in width, laid down
along the [$1\overline{1}0$] direction with lengths of 3.58 nm (Stripe1) and
7.17 nm (Stripe2), were examined. It should be noted that the spin magnetic
moments of the Mn atoms quite generally depend upon the number of nearest
neighbors. Fewer neighbors yield higher spin magnetic moments, with values
ranging from 3.98$\mu_{\text{B}}$/atom, for atoms located at the NW edges, to
3.08$\mu_{\text{B}}$/atom for the most central Mn site on Stripe2. It is also
found here that the Mn atoms induce a relatively small spin polarization on
the W atoms adjacent to the Mn nanostructures, with values $\approx$
0.20$\mu_{\text{B}}$/atom.

Figure 1:  (Color online) Mn nanostructures adsorbed on a W(110) surface:
linear nanowire (NW), double nanowires (DNW), nanostripes (Stripe1 and
Stripe2). The blue (dark) and gray (light) balls indicate Mn and W atoms,
respectively.

Fig. 2 displays the non-collinear ground-state magnetic spin configurations
calculated self-consistently for the nanostructures described above. The
ground state spin moments of the NW exhibits an (ortho) helical-spin-spiral
along the [$1\overline{1}1$] direction, with a periodicity of $\approx$ 1.1
nm. The angles between spin magnetic moments of nearest neighbors (NN) Mn
atoms is $\approx$ 90∘ and the spin chirality is clock-wise. The DNM system
presents a complex non-collinear configuration in which the spin magnetic
moments of NN Mn atoms in different rows are almost parallel, whereas among
those located on the same row it is $\approx$ 135∘. The first-principles
calculations for the nanostripes reveal cycloidal spin spiral ground states,
with long-range modulations along the [$1\overline{1}0$] direction. Stripes 1
and 2 display half-periods of $\approx$ 3.6 nm and $\approx$ 7.24 nm,
respectively, and inter-row antiferromagnetic ordering. These spin
configurations resemble the ones observed by spin-polarized scanning tunneling
microscopy measurements in a monolayer of Mn/W(110), which display a spin-
spiral period of $\approx$ 6 nm propagating along the [$1\overline{1}0$]
direction Bode  _et al._ (2007).

Figure 2:  Non-collinear ground-state spin magnetic configurations (moments
shown by arrows) for Mn nanostructures adsorbed on a W(110) surface. (a)
Nanowire (NW) with 15 Mn atoms; (b) Double nanowire (DNW) with 9 Mn atoms in
length; (c) Stripe1 with with 3 Mn atoms in breadth and 9 Mn atoms in length;
(d) Stripe2 with with 3 Mn atoms in breadth and 17 Mn atoms in length.

The calculated Heisenberg exchange interactions and the modulus of the DM
vectors for the Mn nanostructures are shown in Fig. 3 for several pairs of Mn
atoms as functions of their interatomic distances. We have only included the
values for pairs of Mn atoms located around the center of the nanostructures.
Those referring to atoms near the ends of the nanostructures’ axes do not
differ significantly from the ones shown in the figure and follow essentially
the same qualitative trends.

Figure 3: The bi-linear Heisenberg exchange ($J_{ij}$) (a) and the
Dzyaloshinskii-Moriya strengths ($|\vec{D}_{ij}|$) (b) interactions calculated
for the Mn nanostructures in their ferromagnetic configurations. The
interactions between Mn atoms around the nanostructure central regions are
plotted as functions of the inter atomic Mn distances: black circles refer to
the nanowire (NW); red squares are for the double-nanowire (DNW); green
diamonds stand for the Stripe1 and blue triangles for Stripe2.

We note that $|J_{ij}|$ is larger for nearest neighbors Mn atoms and decreases
relatively fast with the inter atomic distance. The trend of the distance
dependence of the $|J_{ij}|$ parameters is also seen to be similar for all
systems investigated here (Fig. 3a). It should also be noted that positive
(negative) values of $J_{ij}$ between the magnetic moments located at sites
$i$ and $j$ represent a tendency for ferromagnetic (anti-ferromagnetic)
alignment.

As shown in Fig. 3(b) the DM strength ($|\vec{D}_{ij}|$) between Mn NN pairs
is smaller than the NN Heisenberg exchange interaction, and analogously for
further neighbors. The DM vectors for all pairs of Mn atoms in the NW are
along the $[001]$ direction. They are all in the plane, forming an angle of
$\approx$ 54∘ with the chain direction. Since the exchange interaction between
NN Mn atoms is antiferromagnetic, in order to determine the magnetic
anisotropy for the NW, we also performed selfconsistent calculations
considering a collinear ordering, and the magnetic anisotropy energy (MAE) for
the NW was obtained by calculating the energy difference ($\Delta{E}$) between
the cases where the Mn moments are perpendicular to the plane (${E}_{\perp}$)
and in-plane (${E}_{\parallel}$) along the wire direction. We obtained
$\Delta{E}={E}_{\perp}-{E}_{\parallel}$ $\approx$ 1meV, indicating that the
easy axis is along the [$1\overline{1}1$] direction (in the plane). The MAE in
the presently investigated systems is therefore smaller than the NN DM and
Heisenberg interactions, which represents a rather typical situation. The
interplay between the $J_{ij}$ and $|\vec{D}_{ij}|$ explains the helical spin-
spiral obtained for the ground state magnetic configuration of the NW.

It is clear from Fig. 3 that the exchange interactions for the DNW, Stripe1
and Stripe2 nanostructures, have finite, but significant range, and is
oscillatory. The DM vectors for the DNW are in-plane and they all point along
the $[001]$ direction. The modulus of the DM vectors is also finite-ranged.
The non-collinear spin textures shown in Fig. 2 can be explained from a
competition between the DM and the bilinear exchange interactions. Concerning
the stripes (Stripe1 and Stripe 2), the calculated chiral order of the spin-
moments can be understood from the directions of the DM vectors, which point
along the $[001]$ direction, leading to a tendency towards the formation of a
cycloidal spin-spiral state that is propagating along the [$1\overline{1}0$]
direction (which is the stripes axes, see Fig. 1). Moreover, the periods of
these spin-spirals are rather sensitive to changes in the $|\vec{D}_{ij}|$
values, since they are rather similar to the $J_{ij}$ parameters, in both
systems (see Fig. 3).

Let us now turn our attention to the ground state orbital magnetic moment
configurations of these nanostructures. The calculated orbital moments are
depicted by red arrows in Fig. 4. The magnitude of the Mn orbital magnetic
moments is $\approx$ 0.15$\mu_{\text{B}}$/atom, whereas for the W atoms it is
much smaller $\approx$ 0.005$\mu_{\text{B}}$/atom. One may note that the local
spin and orbital magnetic moments of each Mn atom in general are not
collinear, as Fig. 4 illustrates.

Figure 4: (Color online) The spin and orbital ground state magnetic orderings
calculated for Mn nanostructures adsorbed on a W(110) surface: (a) Mn nanowire
(NW); (b) double nanowire (DNW), (c1-c2) top and lateral views of Stripe1, and
(d1) Stripe2. (d2) Moments directions for only one row of the Stripe2 system.
Yellow and blue arrows represent the directions of the local spin magnetic
moment and the red arrows symbolize the corresponding orbital magnetic moment
orientations.

For the NW the orbital and spin moments are at an angle on each Mn site, but
they present very similar spiral patterns, i.e. they have the same
periodicity. In the case of the DNW, the spin and orbital magnetic
configurations also have similar periodicity. However, in case of the stripes,
the orbital moments have a distinct magnetic configuration, which is
drastically different from, and incommensurate with, the spin arrangement.
Except for the edge atoms, the orbital magnetic moments tend to align in-plane
along the stripe’s direction, both for Stripe1 and Stripe2.

In order to analyse the role played by the spin-orbit coupling on the ground
state spin and orbital magnetic configurations, we have explored how the spin-
and orbital moments vary for different values of the spin-orbit coupling
strength ($\lambda$). In Fig. 5 we show results of the NW ground state spin
and orbital magnetic moments arrangements calculated self consistently for
different values of $\lambda_{\eta}$. Here, $\lambda_{\eta}=\eta\lambda$ where
$\lambda$ is the calculated value of SOC strength, for each Mn and W atom,
used in determining the results depicted in Fig. 4.

Figure 5: (Color online) The spin and the orbital magnetic configurations for
Mn nanowire (NW) on a W(110) surface for several spin-orbit coupling values
$\lambda_{\eta}$, were $\eta=1$ refers to 100% of the true SOC value, and e.g.
$\eta=0.1$ one tenth of this value. (a1)-(a3): Mn nanowire (NW) with
$\eta=0.1$, $2$ and $10$, respectively. (b1)-(b2): Stripe1 for $\eta=0.1$ and
$10$. Yellow (red) arrows denote the spin (orbital) moment directions.

We note that the SOC strength has a significant influence on the orientation
dependence of the orbital magnetic orderings as well as the magnitude of the
Mn orbital moments. The latter tends to increase with the enhancement of the
SOC coupling values, with values ranging from $\approx$
0.02$\mu_{\text{B}}$/atom, for $\eta=0.1$, to $\approx$
0.6$\mu_{\text{B}}$/atom for $\eta=2$. For relatively small values of
$\lambda_{\eta}$ the spin and orbital moments of each Mn atom are virtually
uncoupled, whereas for sufficiently large SOC values they tend to align in the
same direction, as expected.

In order to consider a concrete microscopic Hamiltonian, we have considered
the following model:

$\displaystyle\hat{H}=-\sum_{i\neq
j}J_{ij}\hat{\vec{S_{i}}}\cdot\hat{\vec{S_{j}}}-\lambda\sum_{i}\hat{\vec{L_{i}}}\cdot\hat{\vec{S_{i}}}-\hat{H}^{(i)}_{\mathrm{CF}},$
(1)

where $J_{ij}$ denote the effective Heisenberg (isotropic) exchange
interactions between magnetic moments located on sites $i$ and $j$,
$\hat{\vec{S_{i}}}$ and $\hat{\vec{L_{i}}}$ represent the spin and orbital
angular momentum operators, respectively, and $\lambda$ is the spin-orbit
coupling constant. $\hat{H}^{(i)}_{\mathrm{CF}}$ symbolizes the crystal field
Hamiltonian.

From the point of view of a more general approach Secchi  _et al._ (2015) this
means that we neglect orbital-orbital and spin-orbital interactions at
different sites but take them into account explicitly at the same site. The
intrasite orbital-orbital interaction is nothing but the crystal-field
splitting term, as is clear from Stevens operator-equivalent method in the
crystal-field theory Stevens (1952). Neglecting orbital-related intersite
interactions is justified by the smallness of orbital magnetic momenta.

For simplicity we shall neglect the explicit presence of the substrate and
deal with an ideal atomic wire, characterized by $C_{\infty v}$ point group
symmetry. The leading term in the expansion of the crystal field potential for
$C_{\infty v}$ symmetry has the form (see e.g.Mulak and Gajek (2000)):

$\displaystyle\hat{H}^{(i)}_{\mathrm{CF}}=V_{\mathrm{CF}}(3\hat{L}^{2}_{i,z}-\hat{\vec{L_{i}}}^{2}).$
(2)

Here, we assume that the wire is in $z$ direction. Depending on the sign of
$V_{\mathrm{CF}}$, the wire will have easy axis or easy plane for orbital
magnetic moments.

According to the model (1), the long-range order character of the spin
interactions is primarily defined by the values of the interatomic exchange
constants $J_{ij}$. In the Mn chains, for example, the ground state spin
spiral configuration results from competing interatomic interactions. At the
same time, the orbital moment on each atom experiences a combined effect of
spin-orbit coupling, which tends to align it parallel to the spin moment, and
the crystal field that tends to line it up along the orbital easy-axis (or
easy plane) van der Laan (2001). Thus, the final direction of local orbital
angular moment $\vec{L}$ depends upon the relative strength of these two
terms. In what follows we shall show that this simple model captures the main
effects observed in our non-collinear DFT calculations.

The model, defined by Eqs.(1) and (2) was solved for a finite-size chain of
spins. We considered a chain of 20 sites with both spin and orbital moments
treated as classical vectors of unitary length. Since we aimed at describing
$3d$ system, we assume that the $J_{ij}$’s are much larger than $\lambda$.
Thus, the spin configuration is primarily defined by the exchange
interactions. Restricting the number of inter-site interactions to nearest
($J_{1}$) and next-nearest neighbours ($J_{2}$), the ground state spin
configuration is defined by the maximum of the Fourier transformed exchange
interaction:

$\displaystyle J(q)=2J_{1}\cos(q_{z}a)+2J_{2}\cos(2q_{z}a),$ (3)

where $q$ is the wavevector of the spiral and $a$ is the inter atomic
distance.

For our simulations, we set $J_{1}$=1 (ferromagnetic) and $J_{2}$=-1
(antiferromagnetic), which resulted in the cycloidal spiral ground state with
wavevector $q_{\mathrm{GS}}$=0.21$\times\frac{2\pi}{a}$. The parameters
$\lambda$ and $V_{\mathrm{CF}}$ were both positive and their ratio was varied.
The summary of the results is shown in Fig. 6. One can see that in the regime
when $V_{\mathrm{CF}}<\lambda$, spin and orbital moments are nearly parallel
to each other. As $V_{\mathrm{CF}}$ is increasing, the distribution over the
$\theta(=\angle(\vec{L}_{i},\vec{S}_{i}$)) angles gets broader, indicating
that the orbital moments tend to align more and more along the easy axis
direction of the magnetization. Finally, for large $V_{\mathrm{CF}}$, the
directions of the orbital moments become locked and each of $L_{i}$’s is
oriented parallel or antiparallel to $z$ axis, depending on the sign of the
$S_{i,z}$ component.

Figure 6: Left: Histogram of the obtained distribution of the angles
($\theta$) between spin and orbital moments at each site. Right: The ground
state magnetic order. Spin (orbital) moments are shown at violet (orange)
arrows. $Z$ axis is along the direction of the wire.

##  IV Conclusion

In this work we have explored the magnetism of Mn chains of various width (1 -
3 atomic layers) on a W (011) substrate, using a real space electronic
structure method. We find that the spin-moments of all investigated systems
align in a non-collinear way. This is the result of antiferromagnetic nearest
neighbour interactions, coupled to ferromagnetic interactions of longer range.
The orbital magnetism is induced via spin-orbit effects, and we have found
that spin- and orbital moments of a given Mn atom can exhibit a rather large
angle between them, and that the period of the two spin-structures can be
distinctly different. The magnetic chirality of spin- and orbital moments is
discussed in the light of a miroscopic theory that involves Heisenberg
exchange, spin-orbit coupling and crystalline electric field. Such
competitions have been discussed before for bulk materials Nikolaev and
Solovyev (2014); Grånäs  _et al._ (2014). Mn on W(011) represent one more
example of such exotic couplings, and that they are fully possible also in
nano-sized objects.

## Acknowledgements

M. M. B-N., R. B. M. and A. B. K. acknowledge financial support from CAPES and
CNPq, Brazil. The calculations were performed at the computational facilities
of the National Laboratory for Scientific Computing (SDumont, LNCC/MCTI) and
CENAPAD, Brazil. O.E. acknowledges support from the Swedish Research Council
(VR), the Knut and Alice Wallenberg Foundation, the Swedish Foundation for
Strategic Research (SSF), the Swedish Energy Agency (Energimyndigheten), ERC
(synergy grant FASTCORR, project 854843), eSSENCE, and STandUPP. M.I.K.
ackhowledges support from ERC (synergy grant FASTCORR, project 854843). The
work of Y.K. is supported by VR under the project No. 2019-03569.

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