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\texttt{Comments, corrections, and related references welcomed,
as always!}\\[.5em]
{\TeX}ed \today
\vspace{2em}
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\title{Some empty inverse limits}
\thanks{This preprint is readable online at
\url{/~gbergman/papers/unpub}.
}
\subjclass[2010]{Primary: 06A06, 03E04, 18A30.
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Secondary: 08B20, 13C99, 16D99, 20B30.}
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\keywords{empty inverse limit of sets, cofinality, module}
\author{George M. Bergman}

\dedicatory{To the memory of Leon Henkin, 1921-2006}

\begin{abstract}
A simplified proof is given of L.\,Henkin's result that every directed
partially ordered set $I$ of uncountable cofinality indexes a system
of nonempty sets and surjective set maps with empty inverse limit.
Strengthening a result of N.\,Aronszajn, rediscovered by
G.\,Higman and A.\,H.\,Stone, it is then shown
that a large class of such partially ordered sets index
systems with these properties in which the sets occurring are countable.
Examples are also given of groups $G$ admitting $\!\omega_1\!$-indexed
systems of transitive $\!G\!$-sets with empty inverse limits, whence
their group rings $k\,G$ admit systems of cyclic modules with surjective
homomorphisms and zero inverse limits.
Several questions are posed.
\end{abstract}
\maketitle

% - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
\section{Introduction, definitions, and overview.}\label{S.intro}

Recall that a partially ordered set $(I,\leq)$ is said to be
(upward) directed if
%
\begin{equation}\begin{minipage}[c]{35pc}\label{d.dir}
for all $i, j\in I,$ there exists $k\in I$ with $k\geq i$ and $k\geq j,$
\end{minipage}\end{equation}
%
and that an {\em inversely directed system} (or {\em inverse
system}) of sets indexed by the directed partially ordered set
$I$ means a family $(X_i,X_{ji})$ such that
%
\begin{equation}\begin{minipage}[c]{35pc}\label{d.Xi}
for all $i\in I,$ $X_i$ is a set,
\end{minipage}\end{equation}
%
%
\begin{equation}\begin{minipage}[c]{35pc}\label{d.Xij}
for all $i\geq j$ in $I,$ $X_{ji}$ is a set map $X_i\rightarrow X_j,$
\end{minipage}\end{equation}
%
%
\begin{equation}\begin{minipage}[c]{35pc}\label{d.Xii}
for all $i\in I,$ $X_{ii}$ is the identity map of $X_i,$
\end{minipage}\end{equation}
%
%
\begin{equation}\begin{minipage}[c]{35pc}\label{d.Xijk}
for all $i\geq j\geq k$ in $I,$ one has $X_{ki}=X_{kj}\,X_{ji}.$
\end{minipage}\end{equation}
%
In this situation, the {\em inverse limit} of this system is
defined to be the set
%
\begin{equation}\begin{minipage}[c]{35pc}\label{d.Lm}
$\Lm_{i\in I}\ X_i~\ =~\ \{(x_i)_{i\in I}\in\prod_{i\in I}
X_i\ |\ \forall\ i\geq j,\ X_{ji}(x_i)=x_j\}.$
\end{minipage}\end{equation}
%

(I generally prefer to reverse the inequalities
in~(\ref{d.dir})-(\ref{d.Lm}), and thus
index inverse systems by {\em downward} directed sets, since the
standard way to make a partially ordered set into a
category is to have morphisms go from lower to higher elements, and
it is preferable to have functors covariant when there is no
intrinsic reason for the opposite choice.
However, in most of the explicit examples in this note, this would
require introducing the opposites of familiar partial orderings,
leading to awkwardness in interpreting inequality signs.
So, reluctantly, but in conformity with the notation of
\cite{LH} and \cite{H+S}-\cite{DZ}, I am using the above definitions.)

It is well known and easy to show (by compactness, in one form or
another) that the inverse limit of an
inverse system of {\em finite} nonempty sets is always nonempty.
It is also well known that
without the hypothesis of finiteness this fails.
For instance, suppose $S$ is a set and $I$ some family of subsets of $S$
which forms a directed set under inclusion, and we define an
inverse system by letting $X_i=S-i$
(the relative complement of the subset $i$ in $S),$ and for
$i\geq j\in I$ letting $X_{ji}$ be the inclusion of $X_i$ in $X_j.$
We see that $\Lm_I\,X_i$ can be identified with the intersection
$\bigcap_{i\in I} S-i,$ so that if, for instance, $S$ is infinite
and we take $I$ to consist of its finite subsets,
the $X_i$ are all nonempty but have empty inverse limit.
(The above is the one example in this note where it would have been more
convenient to use the opposite of the present convention on
ordering; we could then have taken $I$
to consist of the \emph{cofinite} subsets of $S,$ and $X_i=i.)$

In fact, every directed partially ordered set $I$ with no
greatest element indexes a system of nonempty sets $X_i$ and inclusion
maps $X_{ji}$ with empty inverse limit, defined by taking
$X_i=\{j\in I\ |\ j\geq i\}.$

In the above examples, it is the nonsurjectivity of the $X_{ji}$
that leads to the empty inverse limits, so we might hope for better
behavior in systems with surjective maps.
Henkin \cite{LH} analyzes this situation, and shows that
a directed partially ordered set $I$ has the
property that every inverse system of nonempty sets and surjective
maps indexed by $I$ has nonempty inverse limit if and only
if $I$ has cofinality $\aleph_0$ or $1.$
The hard part of that result is a construction associating
to every $I$ of uncountable cofinality an inverse system with
surjective maps but empty inverse limit; I will give a
simplified version of the construction below.

The sets $X_i$ of that construction (in both
\cite{LH} and my version) are uncountable.
This, and the fact that no such example can occur with finite $X_i,$
leads to the question of whether it can occur with all $X_i$ countable.
Examples have been given showing that this, too, can happen.
Taking them in reverse chronological order,
W.\,Waterhouse, in the $\!4\!$-sentence paper \cite{WCW}, notes that
we can get an example by taking any uncountable set $S,$ letting
$I$ be the set of finite subsets of $S$ ordered by inclusion,
letting $X_i,$ for each $i\in I,$
be the set of one-to-one maps of $i$ into $\omega,$
and letting the  $X_{ji}$ be the restriction maps.
Higman and Stone \cite{H+S} give a more complicated example, recalled
below, with $I=\omega_1,$ the first uncountable ordinal, and show
that the properties of this index set allow one to
obtain from that example inverse systems of nontrivial {\em
groups, rings,} etc., with surjective maps but trivial inverse limits.
I am grateful to P.\,B.\,Levy for pointing out that
Higman and Stone's result, minus the algebraic applications,
was anticipated in the 1930's by the construction of
``Aronszajn trees'' in \cite{GK}, namely, trees (there called
``tableaux ramifi\'{e}s'', i.e., ``branched diagrams'')
of height $\omega_1$ (there denoted
$\Omega)$ in which every element has elements above it at every
height, yet there are no branches of full height $\omega_1.$
(G.\,Kurepa \cite[p.132]{GK} says the construction was
communicated to him by N.\,Aronszajn around 1936.
It is much lengthier than the later construction
of Higman and Stone, and I confess that I have not read it.)

Below, I show that Higman and Stone's method leads to
examples of the same phenomena with a much larger class of
index sets, properly including all directed
partially ordered sets of cofinality $\aleph_1.$
I also note examples showing that in the nontriviality result
for inverse limits of finite nonempty sets, one cannot
replace ``finite set'' with ``transitive $\!G\!$-set''
or ``finitely generated module''; and
I record some questions that our results leave open.

\vspace{6pt}
In its present form, this is a discussion paper.
If I submit it for publication, I probably should drop some of the
commentary, as well as the proof of the Aronszajn-Higman-Stone result
(Theorem~\ref{T.A+H+S}), which does not improve
on Higman and Stone's proof, as well
as any other results that I learn are not new;
and, perhaps, familiar definitions and examples.
I welcome readers' thoughts on what to keep.

\section{Main results.}\label{S.main}
In what follows, we will regard a partially ordered set $(I,\leq)$ as a
category with morphisms going from larger to smaller elements
(the opposite of the standard convention),
so that we can call a system of sets and morphisms as
in~(\ref{d.dir})-(\ref{d.Xijk}) a functor $X: I\rightarrow\Set.$

Recall that for $I$ a partially ordered set, a {\em chain} in $I$
means a totally ordered subset, and a {\em down-set}
means a subset which, if it contains an element, contains all
smaller elements.
By a ``down-subset'' of a subset $x$ of $I,$ we
will mean a subset $y\subseteq x$ which is a down-set
within $x$ (but not necessarily within $I);$
i.e., such that if $i\in y,$ and $j<i$ with $j\in x,$ then $j\in y.$

Here, now, is our simplified version of Henkin's construction.

\begin{theorem}[{\rm Henkin \cite{LH}}\textbf{}]\label{T.LH}
Let $I$ be a directed partially ordered set of
uncountable cofinality.
Then there exists a functor $X:I\rightarrow\Set$ such
that the sets $X_i$ are all nonempty, and the morphisms
$X_{ji}$ are surjective, but $\Lm\,X$ is empty.
\end{theorem}\begin{proof}
For each $i\in I,$ let $X_i$ be the set of all finite
chains $x\subseteq I$ with the property that one and only one
element of $x$ (necessarily the greatest) is $\geq i.$
Given $x\in X_i$ and $j\leq i,$ let $X_{ji}(x)\in X_j$ be
the down-subset of $x$ gotten by dropping from $x$ all
but the smallest of the elements $\geq j.$

It is immediate, without any assumption on the cofinality of $I,$
that the sets $X_i$ are nonempty, and that the $X_{ji}$
satisfy (\ref{d.Xij})-(\ref{d.Xijk}) and are surjective.
Now suppose that $x=(x_i)_{i\in I}\in\Lm_I\,X.$
Since $I$ is directed, for all $i,j\in I$
there exists a common upper bound $k$ for these indices;
it follows that the chains $x_i$
and $x_j$ are down-subsets of the common overchain $x_k.$
Hence one of $x_i,\,x_j$ must be a down-subset of the other;
hence the set of chains
$\{x_i\mid i\in I\}$ is totally ordered by inclusion; hence
their union is a chain $x_\infty.$
In $x_\infty,$ every element belongs to a down-subset $x_i$ which is
finite, hence distinct elements have distinct (finite) numbers
of elements below them, so $x_\infty$ is countable.
Moreover, for every $i\in I,$ the set $x_i,$ and hence also $x_\infty,$
contains an element $\geq i,$ so the countable chain $x_\infty$
is cofinal in $I.$
Thus, the existence of an element $x\in\Lm_I X$ implies that $I$ has
countable cofinality; so if $I$ has uncountable cofinality,
$\Lm_I X$ is empty, as claimed.
\end{proof}

(The converse result, that if $X$ is an inverse system
with surjective maps, indexed by
a directed partially ordered set $I$ of {\em countable} cofinality,
then $\Lm_I\,X$ maps surjectively to each $X_i,$ is
standard and easily proved; cf.\ \cite{LH} and \cite[Lemma~7]{DZ}.)

T.~Kepka (personal communication) calls an inverse system with nonempty
objects and surjective maps but empty inverse limit (or, in the case of
systems of algebras, the same with ``nonempty'' and ``empty''
replaced by ``nontrivial'' and ``trivial'', in the sense appropriate to
the sort of algebra in question,) a ``Henkin spectrum''.
(He uses ``spectrum'' for ``inverse system''.)
Naming the phenomenon after Henkin is appropriate if it proves
important enough to have a name; but my preference is always to
minimize the burden of definitions on the reader, so for now, I will
continue to refer to such objects by stating the properties in question.

Now for the Aronszajn-Higman-Stone result.

\begin{theorem}[{\rm Aronszajn \cite{GK}; proof following Higman and Stone \cite{H+S}}\textbf{}]\label{T.A+H+S}
There exists a functor $Y:\omega_1\rightarrow\Set$ such that the
sets $Y_i$ are all nonempty {\em and countable}, and the morphisms
$Y_{ji}$ are all surjective, but $\Lm\,Y$ is empty.
\end{theorem}\begin{proof}
We start by defining a functor $W:\omega_1\rightarrow\Set$ having
weaker properties.
For each $i\in\omega_1,$ let $W_i$ be
the set of all one-to-one maps $y$ of $i$ into the set $\Q$
of rational numbers, such that the image of $y$
is bounded above in $\Q.$
For $i\geq j,$ let $W_{ji}$ be the restriction map from
functions on $i$ to functions on $j;$ these maps
$W_{ji}$ are easily seen to be surjective; but a member
of their inverse limit would determine a one-to-one
map $\omega_1\to\Q,$ which is impossible, so that inverse limit
is empty.

The sets $W_i$ are uncountable,
but I claim we can find a subfunctor, i.e., a system of subsets
$Y_i\subseteq W_i$ respected by the maps $W_{ji},$ such that each
$Y_i$ is countable, and the restrictions $Y_{ji}$ of the
maps $W_{ji}$ are still surjective; in fact, such that given
any $i>j$ in $\omega_1,$ any $y\in Y_j$ whose image is bounded by a
rational number $r,$ and any $\epsilon>0,$
we can find $y'\in Y_i$ carried by $Y_{ji}$ to $y,$ such
that the image of $y'$ is bounded by $r+\epsilon.$

Indeed, let $i\in\omega_1$ and suppose
subsets $Y_j\subseteq W_j$ have been defined for
all $j<i$ so that the above conditions hold for these subsets.
If $i$ is a successor ordinal $i=j\,{+}\,1,$ then we construct
$Y_i$ simply by choosing, for each $y\in Y_j,$ each rational upper
bound $r$ for the image of $y,$ and each rational number $\epsilon>0,$
an element $y'$ extending $y,$ by sending $j\in i$ to any rational
number in the interval $(r,r+\epsilon);$ and we let $Y_i$ consist
of the countably many elements so chosen.
If $i$ is a limit ordinal, then since it is countable, we can write it
as the supremum of an $\!\omega\!$-indexed chain of smaller ordinals,
$j_0<j_1<\cdots<j_n<\cdots\,.$
Now for every $n\in\omega,$ every $y\in Y_{j_n},$ every rational upper
bound $r$ for the image of $y,$ and every rational number $\epsilon>0,$
let us write $y_n=y,$ then lift this to an element
$y_{n+1}\in Y_{j_{n+1}}$ whose image is bounded above by
$r+\epsilon/2,$ as we may by our inductive hypothesis; then
lift that to an element $y_{n+2}\in Y_{j_{n+2}}$ whose image is
bounded above by $r+3\,\epsilon/4,$ and so forth.
Since each $y_{n+m}$ is an extension of the one before,
their union will be a one-to-one map $y':i\rightarrow \Q$ whose image
is bounded above by $r+\epsilon.$
Making one such construction for each
choice of $y,$ $n,$ $r$ and $\epsilon,$
we get countably many such elements $y',$ and we again
define $Y_i$ to be the set consisting of these.

The resulting subfunctor $Y$ inherits from $W$ the property of having
empty inverse limit, and by construction, the sets $Y_i$ are
countable and the maps between them surjective, as claimed.
\end{proof}

(In \cite{H+S} Higman and Stone used, in place of our $W_i,$ the set of
all {\em isotone} bounded embeddings of $i$ into the {\em real} numbers,
getting emptiness of the inverse limit from the
fact that the real line has countable cofinality.)

We shall now extend this result to systems indexed by certain not
necessarily totally ordered sets, with the help of part~(i) of the next
lemma.
Part~(ii) is recorded for its interest as a partial converse to~(i).
The lemma seems likely to be known; I hope that anyone who recognizes
it will let me know.

\begin{lemma}\label{L.->kappa}
Let $\kappa$ be an infinite cardinal, and $I$ a directed partially
ordered set of cofinality $\kappa.$
Then\\[4pt]
\textup{(i)}~ There exists a surjective
isotone map $I\rightarrow\kappa.$\\[4pt]
\textup{(ii)}~ There does not exist any surjective
isotone map from $I$ to a regular cardinal $>\kappa.$
\end{lemma}\begin{proof}
To show~(i), it suffices to find an isotone
map $f:I\rightarrow\kappa$ whose image has cardinality $\kappa,$
since that image will be order-isomorphic to $\kappa.$

By assumption, there exists a set map $g:\kappa\rightarrow I$
with cofinal image.
For each $i\in I,$ let $f(i)$ be the least $\alpha\in\kappa$
such that $g(\alpha)\geq i.$
The map $f:I\rightarrow\kappa$ is clearly isotone.
Moreover, every $i\in I$ is majorized by $g(f(i)),$ hence
the image set $gf(I)$ is cofinal in $I,$ hence it has cardinality
$\geq\kappa.$
Hence $f(I)\subseteq\kappa$ has cardinality
$\geq\kappa,$ hence equal to $\kappa,$ as required.

To get~(ii), suppose $f$ is an isotone map from $I$ into a regular
cardinal $\lambda>\kappa,$ and $J\subseteq I$ is a cofinal subset of $I$
of cardinality $\kappa.$
Then $f(J),$ being a subset of the regular cardinal $\lambda$ having
cardinality $<\lambda,$ is bounded above by some
$\alpha\in\lambda.$
Since $J$ is cofinal in $I,$ the image under $f$ of any
element of $I$ is also bounded by $\alpha;$ so $f$ is not surjective.
\end{proof}

The above lemma will be used in combination with the next result,
which is well-known in the case where $I$ is a subset of $J$
with the induced ordering, and $f:I\rightarrow J$ is the inclusion;
I would be interested to know whether it has been noted in more
general contexts.
We state this lemma in its natural category-theoretic generality, though
we will only use the case $\C=\Set,$ and, as sketched in the first
paragraph of the proof, the general case actually reduces to that one.

\begin{lemma}\label{L.cofimage}
Let $f:I\rightarrow J$ be an isotone map of directed partially ordered
sets such that $f(I)$ is cofinal in $J,$ and let
$Y:J\rightarrow\C$ be a functor.
Then $\Lm_I\,Y\,{\circ}\,f = \Lm_J\ Y,$ in the sense that if either
limit exists, so does the other, and the natural morphism
$\Lm_J\,Y\rightarrow \Lm_I\ Y\,{\circ}\,f$ is an isomorphism.
\end{lemma}\begin{proof}
By definition, $\Lm_I\,Y\,{\circ}\,f$ and $\Lm_J\,Y$ denote
representing objects for certain contravariant ``set-of-all-cones''
functors on $\C,$ so what must be proved is
that those two functors are isomorphic.
The descriptions of the values of those functors at an object
of $\C$ are equivalent to the concrete description~(\ref{d.Lm}) of the
limits of certain $\!\Set\!$-valued functors on $I,$ respectively $J.$
Examining these, one finds that the general case of the lemma
reduces to the case $\C=\Set;$ so we shall assume $Y$ a
$\!\Set\!$-valued functor.

Now every element $x=(x_j)_{j\in J}\in \Lm_J\,Y$ induces an
element $x\,{\circ}\,f=(x_{f(i)})_{i\in I}\in \Lm_I\,Y\,{\circ}\,f;$
what we have to show is that this map
$-\,{\circ}\,f: \Lm_J\,Y\rightarrow \Lm_I\ Y\,{\circ}\,f$ is bijective.
Given $x=(x_i)_{i\in I}\in \Lm_I\,Y\,{\circ}\,f,$ let us
define $x'\in\Lm_J\,Y$ as follows.
For each $j\in J,$ we can, by hypothesis, find $i\in I$ such
that $f(i)\geq j;$ let $x'_j=Y_{j,f(i)}(x_i).$
To show this well-defined, suppose $f(i)$ and $f(i')$ both majorize $j.$
By directedness of $I,$ we can find $i''\in I$ majorizing both
$i$ and $i'.$
Since $x\in \Lm_I\,Y\,{\circ}\,f,$ we have
$x_i=Y_{f(i),f(i'')}(x_{i''})$ and
$x_{i'}=Y_{f(i'),f(i'')}(x_{i''}),$
so the two candidates for $x'_j$ namely $Y_{j,f(i)}(x_i)$
and $Y_{j,f(i')}(x_{i'}),$ both reduce
to $Y_{j,f(i'')}(x_{i''}),$ and so are equal.
It is straightforward to verify that the
element $x'=(x'_j)_{j\in J}$ indeed
lies in $\Lm_J\,Y,$ and that this construction gives, as
required, a two-sided inverse to
$-\,{\circ}\,f: \Lm_J\,Y\rightarrow \Lm_I\ Y\,{\circ}\,f.$
\end{proof}

We can now get our main result.

\begin{theorem}\label{T.main}
Let $I$ be any directed partially ordered set which admits
an isotone map onto $\omega_1$
\textup{(}for instance, by Lemma~\ref{L.->kappa},
any partially ordered set of cofinality $\omega_1).$
Then there exists a functor $X:I\rightarrow\Set$ such
that the sets $X_i$ are countable and nonempty, and the morphisms
$X_{ji}$ are surjective, but $\Lm\,X=\emptyset.$
\end{theorem}\begin{proof}
Let $X=Y\,{\circ}\,f,$ where $f:I\rightarrow\omega_1$ is a surjective
isotone map, and $Y:\omega_1\rightarrow\Set$ is the functor
of Theorem~\ref{T.A+H+S}.
By Lemma~\ref{L.cofimage}, $\Lm_I\,X=\Lm_{\omega_1}\,Y=\emptyset.$
\end{proof}

\section{Remarks and questions.}\label{S.rem}

The construction of Theorem~\ref{T.A+H+S} begins,
as we saw, with a functor $W$ whose values, like those of
the functor $X$ of Theorem~\ref{T.LH}, are uncountable sets.
What property of $W$ allowed us to ``cut it down'' to a
countable-set-valued functor, while preserving
surjectivity of the connecting maps?
If for each $r\in\Q$ and $i\in\omega_1$ we let $W_i(r)$
denote the subset of $W_i$ consisting of those maps whose images
are bounded above by $r,$ we can express the property we
used as saying that $W$ is the union of a $\!\Q\!$-indexed chain of
subfunctors $W(r),$ such that
%
\begin{equation}\begin{minipage}[c]{35pc}\label{d.W_js->}
Whenever $i\in\omega_1$ is the supremum
of an $\!\omega\!$-indexed chain $j_0<j_1<\cdots<j_n<\cdots\,,$ and
$s\in\Q$ is the supremum of an $\!\omega\!$-indexed sequence
$r_0<r_1<\cdots<r_n<\cdots,$ then any element of
$\Lm_{n\in\omega}\,W_{j_n}(s)$ whose image in each
$W_{j_n}$ lies in $W_{j_n}(r_n)$ arises from an element of $W_i(s).$
\end{minipage}\end{equation}
%

If we had started with a functor on $\omega_1$ with the simpler property
%
\begin{equation}\begin{minipage}[c]{35pc}\label{d.Cj->}
Whenever $i\in\omega_1$ is the supremum of an
increasing chain $j_0<j_1<\cdots<j_n<\cdots\,,$ the natural map
$W_i\rightarrow \Lm_{n\in\omega}\,W_{j_n}$ is surjective,
\end{minipage}\end{equation}
%
this would have been too strong: It would have implied that the
maps $\Lm_{\omega_1}\,W \rightarrow W_i$ $(i\in\omega_1)$ were all
surjective, by essentially the same ``successive lifting'' argument
that shows that for any set-valued functor $X$ with
surjective connecting maps on a directed partially ordered set of
countable cofinality, the maps $\Lm\,X\rightarrow X_i$ are surjective,
using~(\ref{d.Cj->}) to carry us past the limit ordinals.
So $\Lm_{\omega_1}\,W$ would not have been empty, which was one of the
key ingredients of the construction.
The weaker statement~(\ref{d.W_js->}) allowed us to lift elements past
any countable number of limit steps, without forcing the same sort
of lifting to work through all $\aleph_1$ steps.

In the system $X$ of Theorem~\ref{T.LH}, on the other hand, there is
no evident way to find an analog of the family of subfunctors $W(r).$
So we ask

\begin{question}\label{Q.subLH}
Does the functor $X$ given by the $I=\omega_1$ case of the
proof of Theorem~\ref{T.LH} have a subfunctor $Y$ with all $Y_i$
countable and nonempty, and all $Y_{ji}$ surjective?
Does it in fact have a $\!\Q\!$-indexed chain of
subfunctors satisfying~\textup{(\ref{d.W_js->})}?
\end{question}

It would also be interesting to know whether in Theorem~\ref{T.A+H+S},
and hence in Theorem~\ref{T.main}, one can replace $\omega_1$ with
other uncountable regular cardinals.
I twice believed I could prove that theorem with
$\omega_1$ replaced by any infinite successor cardinal $\kappa^+,$
and ``countable set'' replaced by ``set of cardinality $\leq\kappa$''.
The idea was to replace the set $\Q$ by a totally
ordered set $Q_\kappa$ having cardinality $\kappa$ and no
greatest element, such that every
open interval in $Q_\kappa$ had a subset order-isomorphic to $\kappa.$
(This is not hard to construct.)
If for each $i\in\kappa^+$ one then lets $W_i$ denote the set of maps
$i\rightarrow Q_\kappa$ whose image is bounded above in $Q_\kappa$ and
if one defines the subsets $W_i(r)$ $(r\in Q_\kappa)$
as above, one does get the analog
of~(\ref{d.W_js->}) for sequences indexed by cardinals $\leq\kappa.$
However, if one attempts to construct subsets $Y_i$ by a recursive
process analogous to that used in the proof of Theorem~\ref{T.A+H+S},
a problem arises whenever $i$ has uncountable cofinality.
The analog of the $\!\omega\!$-indexed chain $j_0<j_1<\cdots<j_n<\cdots$
with supremum $i$ will be a chain indexed by $\mathrm{card}(i),$ and
the process of lifting our element through this chain is not as easy
as before: there are uncountably many limit steps along the way.
Though at each such step we can lift a sequence of previous choices
to get an element of $W_j(r_j),$ I see no way of assuring that this
limit could be taken to lie in $Y_j(r_j).$
So we ask

\begin{question}\label{Q.kappa}
Does the analog of Theorem~\ref{T.A+H+S} hold if one replaces countable
sets, and $\omega_1,$ by sets of cardinality $\leq\kappa,$ and
$\kappa^+,$ for an arbitrary infinite cardinal $\kappa,$ or at least for
{\em some} such cardinals other than $\omega$?
Can we perhaps even replace $\omega_1$ by other regular cardinals
{\em without} weakening the condition ``countable''?
\end{question}

The directed sets to which Theorem~\ref{T.main} is applicable as
it stands, namely those admitting isotone maps onto $\omega_1,$
already form a much larger class than those of cofinality $\omega_1.$
An easy class of examples with cofinalities
$>\omega_1$ is gotten by taking the direct
product of $\omega_1$ with any directed set of larger cofinality,
under the product ordering (componentwise inequality).
For a less obvious case, let $\kappa$ be any cardinal $>\omega_1,$
and $I$ the partially ordered set of all countable subsets of
$\kappa,$ ordered by inclusion.
The cofinality of $I$ is at least $\kappa$ (since if we merely seek to
majorize the singleton elements of $I,$ no element of $I$ majorizes more
than countably many of them, hence at least $\kappa$ elements
are needed).
However, for any $i\in I,$ the order-type of $i$ as a subset of $\kappa$
is the order-type of a unique element $f(i)\in\omega_1;$
and this function $f$ is easily seen to be an isotone surjection
$I\rightarrow\omega_1.$

\section{Inverse limits of algebras.}\label{S.alg}

Let $\bf V$ be a variety of algebras in the sense of universal
algebra, with all operations finitary, and let
%
\begin{equation}\begin{minipage}[c]{35pc}\label{d.F}
$F:\ \Set\ \to\ \V$
\end{minipage}\end{equation}
%
be the free $\!\V\!$-algebra functor.
Assume that
%
\begin{equation}\begin{minipage}[c]{35pc}\label{d.zeroary}
if $\bf V$
has any constant derived operations, it has zeroary operations.
\end{minipage}\end{equation}

In this situation, we can associate to each element $a$ of a
free algebra $F(S)$ its ``support'',
%
\begin{equation}\begin{minipage}[c]{35pc}\label{d.supp}
$\supp(a)\ =$ the least subset $A\subseteq S$
such that $a\in F(A)\subseteq F(S),$
\end{minipage}\end{equation}
%
a finite subset of $S.$

This operator has the rather eccentric property that
given a set map $u:S\rightarrow T$ and an element
$a\in F(S),$ the set $u(\supp(a))$ is not a
subset of $\supp(F(u)(a)),$ but a superset thereof,
%
\begin{equation}\begin{minipage}[c]{35pc}\label{d.supseteq}
$u(\supp(a))\ \supseteq\ \supp(F(u)(a)).$

\end{minipage}\end{equation}
%
This inclusion can be proper,
since elements of $u(\supp(a))$ may ``vanish'' from
$\supp(F(u)(a))$ as a result of cancellations that
occur when elements of $S$ fall together under $u.$
Hence, given an inverse system $X$ in $\Set$ and an
element $a=(a_i)_{i\in I}\in\Lm_I\,F(X_i),$ it will not in
general be true that
the finite sets $\supp(a_i)\subseteq F(X_i)$ $(i\in\omega_1)$
form a subfunctor of $X.$
If it were, then in the case where $\Lm_I\,X_i=\emptyset,$ we could
deduce that $\Lm_I\,F(X_i)$ was the algebra $F(\emptyset),$ thus
getting ``trivial'' algebras as inverse limits of free algebras subject
to the same sorts of conditions under which we got
$\emptyset$ as an inverse limit of sets.

Nevertheless, Higman and Stone \cite{H+S} note that
if $X$ is an inverse system of sets \emph{indexed by $\omega_1$}
and having empty inverse limit, and we apply $F$ to it, we do get
an inverse system of algebras with inverse limit $F(\emptyset).$
To see this, consider, more generally, any inverse system $X$
of sets whose index set $I$ has the property that every
countable subset of $I$ has an upper bound in $I,$
and suppose $a\in\Lm F(X_i).$
If the set of integers $\mathrm{card(supp}(a_i))$ $(i\in I)$ were
unbounded, then letting $j$ be an upper bound for some countable
set on which they assumed arbitrarily large values, we would get
a contradiction to the statement that $\supp(a_j)$ is finite.
Hence those cardinalities must, rather, assume a maximum value at some
$i_0\in I.$
We then see from~(\ref{d.supseteq}) that on
the set $\{i\mid i\geq i_0\}\subseteq I,$ the maps $X_{ij}$
$(i_0\leq i\leq j)$
carry $\supp(a_j)$ to $\supp(a_i)$ bijectively,
and it follows that every element of $\supp(a_{i_0})$ induces
an element of $\Lm_I\,X_i.$
We deduce
that if that limit is empty, $\Lm_I\,F(X_i)$ is indeed $F(\emptyset).$

Combining this construction with Lemma~\ref{L.cofimage}, we get

\begin{corollary}\label{C.trivalg}
If $I$ is a directed partially ordered set admitting an isotone map
onto $\omega_1,$ and $\mathbf V$ a variety of finitary algebras
satisfying~\textup{(\ref{d.zeroary})},
then there exists a functor $X:I\rightarrow\mathbf V$ such that the
$X_i$ are free algebras on nonempty countable sets, and the
homomorphisms $X_{ji}$ are surjective, but $\Lm\,X$ is the free
algebra on the empty set.\qed
\end{corollary}

This leaves unanswered

\begin{question}\label{Q.F(es)}
Suppose $I$ is a directed partially ordered set of
uncountable cofinality \textup{(}but not assumed to have
upper bounds on countable subsets\textup{)},
and $\mathbf V$ a variety of finitary algebras
satisfying~\textup{(\ref{d.zeroary})}.

\textup{(i)}~If $X: I\rightarrow\Set$ is a functor having
empty inverse limit,
and $F:\Set\rightarrow\mathbf V$ is the free $\!\V\!$-algebra functor,
must $\Lm_I\,F\,{\circ}\,X\cong F(\emptyset)$?

If the answer to \textup{(i)} is negative, we may still ask,

\textup{(ii)}~Does there exist some functor $X: I\rightarrow\mathbf V$
such that the
connecting morphisms $X_{ji}$ are all surjective, but the natural
maps from $\Lm\,X$ to the algebras $X_i$ are not all surjective?
\end{question}

One approach I thought might be applicable to (i) above is
the following.
Assume we have an element $a\in\Lm\,F(X_i)$ whose image in
some $F(X_{i_0})$ does not lie in the subalgebra $F(\emptyset).$
Then let us associate to every $i\geq i_0$ the finite nonempty set
given by the support of the image of $a$ in $F(X_i),$ and try to find a
subsystem of this system of finite sets consisting of singletons,
thus obtaining an element of $\Lm\,X.$

Unfortunately, there may not be such a subsystem -- even
if $I$ is finite!
For instance, let $\V$ be the category of groups, let
$I$ be the $\!4\!$-element lattice $2\times 2,$ let
the set associated with the top element be
$\{x,\,y,\,z\},$ and in the free group on that set, let $a=xy^{-1}z.$
Let the sets associated with the two elements at
the middle level of $I$ be obtained from the top set by the
identifications $x=y$ and $y=z$ respectively, and let the bottom
set be obtained by identifying all three generators.
Then we see that
the support of $a$ is $\{x,\,y,\,z\},$ while we find that
the supports of its images in the other three groups are the singletons
$\{z\},$ $\{x\},$ and $\{x=y=z\};$ so there is no element
of $\supp(a)$ that is mapped to members of all these sets.

(In this example, the ``disappearance'' of generators from
our support-sets was based on those generators falling
together with other generators.
But an element can also
disappear because two \emph{different} elements fall together.
E.g., taking $xyz^{-1}x^{-1}y$ in the free group on $\{x,\,y,\,z\},$
if $y$ and $z$ fall together, then $x$ is lost from the support.)

Here is an example of an inverse system of sets with
surjective maps and empty inverse limit, together with
a family of finite subsets of the members of this inverse system,
which we might try to realize in this way as supports of elements of
free algebras.
Let us take any uncountable set $S,$ let $I$ consist of
the finite subsets of $S,$ and, as in \cite{WCW}, let each $X_i$
$(i\in I)$ be the set of one-to-one maps from $i$ to $\omega.$
Within each $X_i,$ let $Y_i$ be the finite
set consisting of those maps whose image is an initial segment
of $\omega$ (necessarily, the natural number $\mathrm{card}(i)).$
Then the maps $X_j\to X_i$ indeed take each $Y_j$ to a
superset of $Y_i.$
This example itself can't be realized as
desired, because when $j$ contains just one more element than $i,$ the
map $X_j\to X_i$ happens to be one-to-one on elements having images
in $Y_j;$ so we have elements ``vanishing'' without
any elements ``falling together'', which doesn't happen to
supports of elements of free algebras.
However, by slight modifications of this example -- e.g.,
restricting $I$ to sets $i$ of even cardinality; or keeping $I$
as above, but weakening the restriction defining the elements of
$Y_i$ to say that their images lie in the initial segment of $\omega$
of cardinality $2\,\mathrm{card}(i)$ --
one gets cases for which it seems plausible that the system
could be so realized.

\section{$\!G\!$-sets and $\!R\!$-modules.}\label{S.modules}
Tom\'a\v{s} Kepka (personal communication) has asked whether an
inverse limit of nonzero \emph{finitely generated} modules and
surjective homomorphisms can be the zero module.
We shall obtain examples of this by first constructing
analogous examples for group actions.

Recall that a $\!G\!$-set is \emph{transitive} if and only
it is generated by a single element.

\begin{lemma}\label{L.tr_G-set}
There exists a group $G$ and an inversely directed system of
transitive $\!G\!$-sets and surjective $\!G\!$-set homomorphisms,
indexed by $\omega_1,$ having empty inverse limit.
Moreover \textup{(}with considerable additional
work\textup{),} $G$ can be taken abelian.
\end{lemma}

\begin{proof}
{\em Example 1, nonabelian, easy to describe.}
We start with a variant of the set-theoretic construction of
\cite{WCW}: Let $X:\omega_1\rightarrow\Set$ carry
every $i\in\omega_1$ to the set of all one-to-one functions
$i\rightarrow\omega$ whose images are not cofinite, and let
the $X_{ji}$ $(i\geq j)$ be the restriction maps.
Clearly, this set-valued functor again has surjective
maps $X_{ji},$ but empty inverse limit.

Now let $G$ be the symmetric group on $\omega$ (the group of
\emph{all} permutations of that set).
Then $G$ acts naturally on the left on each $X_i,$
making it a transitive $\!G\!$-set, and these
$\!G\!$-set structures are respected by the maps $X_{ji};$ so $X$ can
be regarded as an inverse system of cyclic $\!G\!$-sets with
empty inverse limit.\vspace{.3em}

{\em Example 2, with $G$ abelian.}
Here we will obtain our $G$ as a subgroup of a larger
abelian group, namely
%
\begin{equation}\begin{minipage}[c]{35pc}\label{d.H}
$H\ =\ \Z^{\omega_1\times\omega_1}.$
\end{minipage}\end{equation}
%
Let us define elements $c_{ij}\in H$ for all $i,j\in\omega_1$ by
%
\begin{equation}\begin{minipage}[c]{35pc}\label{d.cij}
$c_{ij}(i',j')\ =\ \begin{cases}
1\quad\mbox{if}\ \ i'<i\ \ \mbox{and}\ \ j'<j,\\[.2em]
0\quad\mbox{otherwise.} &
\end{cases}$
\end{minipage}\end{equation}
%
(The set at which $c_{ij}$ has the value $1$ can be pictured as a
``rectangle''; but note that the height and width of this rectangle,
though countable, are not, in general, finite.)

We now define
%
\begin{equation}\begin{minipage}[c]{35pc}\label{d.G}
$G\ =\ \mbox{the subgroup of $H$ generated by }
\{c_{ji}-c_{ii}\mid i<j\in\omega_1\}.$
\end{minipage}\end{equation}
%
Thus, the support of each generator $c_{ji}-c_{ii}$ is a rectangle
with upper left corner at $(i,i-1)$ (just below the diagonal),
upper right corner at $(j-1,i-1),$
and lower corners at $(i,0)$ and $(j-1,0).$

By a ``cross-section'' of an element $h\in H,$ we shall mean the
element of $\Z^{\omega_1}$ gotten, in the obvious way, from the
coordinates of $h$ by fixing any $j\in\omega_1,$ and letting $i$ vary.
We see that
%
\begin{equation}\begin{minipage}[c]{35pc}\label{d.few_sections}
every element of $G$ has only finitely many distinct cross-sections,
\end{minipage}\end{equation}
%
\begin{equation}\begin{minipage}[c]{35pc}\label{d.bounded}
every cross-section of an element of $G$ has bounded support; i.e., its
$\!i\!$-th coordinate is $0$ for all sufficiently large~$i\in\omega_1,$
\end{minipage}\end{equation}
%
\begin{equation}\begin{minipage}[c]{35pc}\label{d.0_above_diag}
all elements of $G$ have value zero on and above the diagonal
of $\omega_1\times\omega_1.$
\end{minipage}\end{equation}
%
(The construction below would work if we instead
took $G$ to consist of all elements
of $H$ satisfying the conclusions
of~(\ref{d.few_sections})-(\ref{d.0_above_diag});
or any group between the group defined in~(\ref{d.G})
and that one.)

For each $i\in\omega_1,$ let us now define
%
\begin{equation}\begin{minipage}[c]{35pc}\label{d.Hi}
$H_i\ =\ \Z^{\omega_1\times i}.$
\end{minipage}\end{equation}
%
Thus we have obvious coordinate-dropping homomorphisms,
%
\begin{equation}\begin{minipage}[c]{35pc}\label{d.fi}
$f_i: H\to H_i,\quad f_{ij}: H_j\to H_i\ \ (i\leq j).$
\end{minipage}\end{equation}
%
The homomorphisms $f_i$ allow us to regard each
$H_i$ as an $\!H\!$-set, and hence as a $\!G\!$-set.
We now define, within each $H_i,$ the $\!G\!$-orbit
(i.e., coset of $f_i(G)),$
%
\begin{equation}\begin{minipage}[c]{35pc}\label{d.Xi=}
$X_i\ =\ f_i(c_{ii}+G)$\quad(see~(\ref{d.cij})).
\end{minipage}\end{equation}
%
Clearly all elements of $X_i$ satisfy
the conclusions of~(\ref{d.few_sections}) and~(\ref{d.bounded})
(with ``cross-section'' defined for elements of $H_i$ as for
elements of $H);$ while rather than~(\ref{d.0_above_diag}), we see that
they satisfy
%
\begin{equation}\begin{minipage}[c]{35pc}\label{d.1_above_diag}
all elements of $f_i(c_{ii}+G)$ have value $1$
on and above the diagonal.
\end{minipage}\end{equation}

I claim now that for $i\leq j,$
the map $f_{ij}:H_j\to H_i$ carries the orbit $X_j$ to the orbit $X_i.$
Indeed, $f_{ij}(f_j(c_{jj}))=f_i(c_{jj}),$ and since
$f_i$ discards coordinates with
second subscript $\geq i,$  this equals $f_i(c_{ji}).$
Since $c_{ji}-c_{ii}\in G,$ the $\!G\!$-orbit of this element
contains $f_i(c_{ii}),$ and so equals $X_i.$

Note that $\Lm_{\omega_1}\,H_i=H,$ hence that
$\Lm_{\omega_1}\,X_i$ can be identified with a subset of $H.$
Suppose that subset were nonempty;
say $y\in\Lm_{\omega_1}\,X_i.$
Then as a function on $\omega_1\times\omega_1,$ $y$
inherits from the $X_i$ the conclusions of~(\ref{d.bounded})
and~(\ref{d.1_above_diag}), while
the property~(\ref{d.few_sections}) of the $X_i$
implies that below each horizontal line
in $\omega_1\times\omega_1,$ $y$ has only finitely
many distinct cross-sections.
But since the ordinal $\omega_1$ has uncountable cofinality, this
implies that $y$ has only finitely many cross-sections altogether;
i.e., does in fact satisfy~(\ref{d.few_sections}).
Since each of these finitely many
cross-sections has bounded support, there
is some $i_0\in\omega_1$ containing all their supports.
Applying this fact to the $\!i_0\!$-th cross-section, we see
that the $(i_0,i_0)$ coordinate of $y$ must be $0,$
contradicting~(\ref{d.1_above_diag}), and completing the proof
that the system of $\!G\!$-sets $X$ has empty inverse limit.
\end{proof}

We mention some easy variants of the above examples.
In the noncommutative example we could (in the spirit of
Higman and Stone's original proof of Theorem~\ref{T.A+H+S})
let $X:\omega_1\rightarrow\Set$ take each $i\in\omega_1$
to the set of \emph{order}-embeddings of $i$ in $\Q$ (or $\R)$
whose image is bounded above, and let $G$ be the group
of all order-automorphisms of $\Q$ (respectively $\R).$
(Or, if we
wish, the subgroup thereof consisting of those order-automorphisms
that act as the identity on all sufficiently large arguments.)
In the commutative example, we could, of course, replace
the $\Z$ in $\Z^{\omega_1\times\omega_1}$ by any nonzero abelian
group $A,$ and the $1$ in~(\ref{d.cij}) by any nonzero element of $A.$

To get the module-theoretic result Kepka asked for, let
$G$ and $X$ be as in the statement of the above lemma,
let $k$ be any field, and let $F$ be the free $\!k\!$-module functor,
which associates to every set a vector space with that set as basis.
Then $F\,{\circ}\,X$ gives a functor from $\omega_1$ to
$\!k\!$-vector-spaces, with surjective connecting maps, which
(by the result of Higman and Stone by which we led up to
Corollary~\ref{C.trivalg}) has the zero vector space as inverse limit.

Combining the action of $G$ on $X$ with the vector-space-valued
functor $F,$ we see that $F\,{\circ}\,X$ can be made a functor
from $\omega_1$ to left modules over the group algebra $k\,G.$
For each $i,$ any element of $X_i$ generates $F(X_i)$ as a
$\!k\,G\!$-module, so these modules are cyclic.
Taking $R=k\,G,$ we get

\begin{corollary}\label{C.cyclic_mdl}
There exists a ring $R,$ and an inversely directed system of
cyclic \textup{(}in particular, finitely generated\textup{)}
nonzero left $\!R\!$-modules, and surjective $\!R\!$-module
homomorphisms, whose inverse limit is the zero $\!R\!$-module.
In fact, $R$ can be taken commutative.\qed
\end{corollary}

From examples of this phenomenon for one ring, group, etc., one can
get such examples for others: If a ring $R$ admits an inverse system
$X$ as in Corollary~\ref{C.cyclic_mdl}, and $S$ is an over-ring which
is free as a \emph{right} $\!R\!$-module, then applying extension
of scalars to $X,$ one finds that $S$ also admits such a system.
The analogous statement for a monoid and overmonoid is likewise clear;
applying this fact about monoids to a group $G,$ we see
that if $G$ has the property of Lemma~\ref{L.tr_G-set}
then {\em every} overgroup inherits that property.
The property is also clearly inherited by rings, groups or
monoids that admit homomorphisms {\em onto} rings, groups or
monoids with the property; so in particular, all free rings, free
groups and free monoids of large enough rank have the property.
Finally, if a ring, group or monoid $R,$ $G$ or $M$ has a
subring, subgroup, or submonoid $R',$ $G'$ or $M'$ such that
$R,$ $G$ or $M$ is finitely generated as a left
$\!R'\!$-module, $\!G'\!$-set or $\!M'\!$-set, then every
finitely generated left $\!R\!$-module, $\!G\!$-set or $\!M\!$-set
remains finitely generated over $R',$ $G'$ or $M',$ so the property
carries over from $R,$ $G$ or $M$ to $R',$ $G'$ or $M'.$

\begin{question}\label{Q.whichrings}
Can a {\em countable} ring $R$ admit an inverse system of
finitely generated nonzero left $\!R\!$-modules and surjective
module homomorphisms with zero inverse limit?

Can a countable monoid $M$ admit an inverse
system of finitely generated nonempty left $\!M\!$-sets and surjective
$\!M\!$-set homomorphisms with empty inverse limit?

Can a countable group $G$ admit an inverse
system of transitive left $\!G\!$-sets and
$\!G\!$-set homomorphisms with empty inverse limit?
\end{question}

Other examples of trivial module-theoretic inverse limits
are obtained in \cite[\S7-\S8]{F+G+S}.

In \cite[Theorem~4]{lim_inj}, we use the result of
Aronszajn-Higman-Stone (Theorem~\ref{T.A+H+S} above)
in a slightly more subtle fashion than here, to get
inverse systems of modules and surjective homomorphisms
with a different contrast
between the properties of the given modules and of their inverse limit.

\begin{thebibliography}{1}

\bibitem{lim_inj} George M. Bergman,
{\em Every module is an inverse limit of injectives,} preprint,
4\,pp., readable at \url{/~gbergman/papers}.

\bibitem{F+G+S} L\'{a}szl\'{o} Fuchs, R\"{u}diger G\"{o}bel
and Luigi Salce,
{\em On inverse-direct systems of modules,}
J. Pure Appl. Algebra {\bf 214} (2010) 322--331.
~MR~\textbf{2010m}:13014.

\bibitem{LH} Leon Henkin,
\textit{A problem on inverse mapping systems},
Proc. Amer. Math. Soc. {\bf 1} (1950) 224--225.
~MR~\textbf{11},\,675e.

\bibitem{GK} Georges Kurepa,
{\em Ensembles lin\'{e}aires et une classe de tableaux
ramifi\'{e}s \textup{(}tableaux ramifi\'{e}s
de M.\,Aronszajn\textup{)}},
Publ. Math. Univ. Belgrade, {\bf 6} (1936) 129-160.
\url{https://elib.mi.sanu.ac.rs/files/journals/publ/6/14.pdf}\,.

% thesis:
% Ensembles ordonn\'{e}s et ramifi\'{e}s.
% \url{https://elib.mi.sanu.ac.rs/files/journals/publ/4/1.pdf}
% online file is missing p.51
% Kurepa says in \cite[132^{II}]{GK} that oE error on pp.96-7

\bibitem{H+S} Graham Higman and A. H. Stone,
\textit{On inverse systems with trivial limits},
J. London Math.  Soc. {\bf 29} (1954) 233--236.
~MR~\textbf{15},\,773b.

\bibitem{WCW} William C. Waterhouse,
\textit{An empty inverse limit},
Proc. Amer. Math. Soc. {\bf 36} (1972) 618.
~MR~\textbf{46}\#8158.

\bibitem{DZ} Daniel Zelinsky,
\textit{Rings with ideal nuclei},
Duke Math. J. {\bf 18} (1951) 431--442.
~MR~\textbf{12}:795e.
% end of 1st parag of sec 2: asks for any nonsurjective example
% Lemma 7: countable cofinality => surjectivity

\end{thebibliography}
\end{document}
