1 Introduction
In [Reference Šobot11], Šobot investigated generalisations of the congruence relation
$a\equiv _n b$
from
${\mathbb Z}$
to its Stone–Čech compactification
$\beta {\mathbb Z}$
, equipped with the usual extensions
$\oplus $
,
$\odot $
of the sum and product of integers. For each
$w\in \beta \mathbb N$
, he introduced a congruence relation
$\equiv _w$
and a strong congruence relation
$\mathrel {\equiv ^{\mathrm {s}}_{w}}$
. In this paper we investigate these notions and prove that, for some
$w$
, the former fails to be an equivalence relation, thereby answering [Reference Šobot11, Question 7.1] in the negative. In fact, we fully characterise those
$w$
for which this happens, and compute the relative quotient when it does not.
Almost by definition,
$u\mathrel {\equiv ^{\mathrm {s}}_{w}} v$
holds if and only if, whenever
$(d,a,b)$
is an ordered triple of nonstandard integers which generate
$w\otimes u\otimes v$
, we have
$d\mid a-b$
. It was proven in [Reference Šobot11] that
$\mathrel {\equiv ^{\mathrm {s}}_{w}}$
is always an equivalence relation, and in fact a congruence with respect to
$\oplus $
and
$\odot $
but, perhaps counterintuitively for a notion of congruence, there are some ultrafilters
$w$
for which
$w\mathrel {\not \equiv ^{\mathrm {s}}_{w}} 0$
. On the other hand, the relation
$\equiv _w$
does always satisfy
$w\equiv _w 0$
but, as we said above, it may fail to be an equivalence relation. So, in a sense, these two relations have complementary drawbacks, and it is natural to ask for which ultrafilters
$w$
these drawbacks disappear. Our main result says that
$\equiv _w$
is well-behaved if and only if
$\mathrel {\equiv ^{\mathrm {s}}_{w}}$
is, if and only if the two relations collapse onto each other. This is moreover equivalent to the quotient
$(\beta {\mathbb Z}, \oplus )/\mathord {\mathrel {\equiv ^{\mathrm {s}}_{w}}}$
being a profinite group, which in fact can be explicitly computed. More precisely, if we denote by
$\mathbb P$
the set of prime natural numbers and by
${\mathbb Z}_p$
the additive group of
$ p$
-adic integers, our main results can be summarised as follows.
Main Theorem (Theorems 4.10 and 7.7).
For every
$w\in \beta \mathbb N$
the following are equivalent.
-
(1) We have
$w\mathrel {\equiv ^{\mathrm {s}}_{w}}0$
. -
(2) The relation
$\equiv _w$
is an equivalence relation. -
(3) The relations
$\equiv _w$
and
$\mathrel {\equiv ^{\mathrm {s}}_{w}}$
coincide. -
(4) The quotient
$(\beta {\mathbb Z}, \oplus )/\mathord {\mathrel {\equiv ^{\mathrm {s}}_{w}}}$
is isomorphic to
$\prod _{p \in \mathbb P} G_{p,w}$
, where
$$\begin{align*}G_{p,w}=\begin{cases} \mathbb{Z}/ p^n {\mathbb Z}, & \text{if} \ n=\max\{k\in\mathbb{N}\cup\{0\}: p^k {\mathbb Z}\in w\} \text{ exists}\,\\ \mathbb{Z}_{p}, & \text{otherwise}.\end{cases}\end{align*}$$
-
(5) The quotient
$(\beta {\mathbb Z}, \oplus )/\mathord {\mathrel {\equiv ^{\mathrm {s}}_{w}}}$
, when equipped with the quotient topology, is a profinite group.
In fact, many more characterisations of those
$w$
satisfying the above equivalent conditions are possible (Theorem 7.7). We believe this to be an indication that these ultrafilters, which we dub self-divisible, are objects of interest, and we study them at length throughout the paper.
In more detail, after briefly recalling the context in Section 2, we prove in Section 3 that
$\equiv _w$
is an equivalence relation if and only if it is transitive and provide an example of an ultrafilter
$w$
such that
$\equiv _w$
is not symmetric. Section 4 is devoted to proving the equivalence of (1) to (3) in the Main Theorem. In Section 5 we provide examples of self-divisible ultrafilters, and study the topological properties of their space in Section 6. In Section 7 we study the quotients
$\beta {\mathbb Z}/\mathord {\mathrel {\equiv ^{\mathrm {s}}_{w}}}$
, and show that each of them may be identified with a quotient of the profinite completion
$\hat {{\mathbb Z}}$
of the integers which embeds in the ultraproduct
$\prod _w {\mathbb Z}/n {\mathbb Z}$
. In the same section, we obtain several further equivalent definitions of self-divisibility, completing the proof of the Main Theorem. We conclude in Section 8 with some final remarks, further directions, and an open problem.
2 Preliminaries
The letters
$u,v,w,t$
will usually denote elements of
$\beta {\mathbb Z}$
, while
$p,q,r$
will typically stand for prime natural numbers. We identify each integer with the corresponding principal ultrafilter. If
$u\in \beta {\mathbb Z}$
we write
$-u$
for
$\{-A: A\in u\}$
and
$u\ominus v$
for
$u\oplus (-v)$
. We extend the usual conventions about usage of
$+$
,
$-$
to
$\oplus $
,
$\ominus $
, e.g., whenever we write
$-u\oplus v$
we mean
$(-u)\oplus v$
, and
$u\ominus v\ominus w$
is to be parsed as
$u\oplus (-v)\oplus (-w)$
. If
$A\subseteq {\mathbb Z}$
, then
$A^{\mathsf {c}}$
denotes
${\mathbb Z}\setminus A$
, and
$\overline A$
denotes the closure of A in
$\beta {\mathbb Z}$
, that is,
$\{u\in \beta {\mathbb Z} : A\in u\}$
. We convene that
$0\notin \mathbb N$
, and use
$\omega $
for
$\mathbb N\cup \{0\}$
.
We adopt some conventions and notations of model-theoretic flavour; some standard references are [Reference Hodges3, Reference Poizat8, Reference Tent and Ziegler13]. Namely, we work in a
$\kappa $
-saturated elementary extension
${}^{\ast }{\mathbb Z}$
of
${\mathbb Z}$
, where the latter is equipped with a symbol for every subset of every cartesian power
${\mathbb Z}^k$
, and where
$\kappa $
is a large enough cardinal, for instance,
$\kappa =(2^{\aleph _0})^+$
. The results obtained do not depend on the particular elementary extension chosen. Moreover, we write
$a\models u$
, or
$u=\textrm {tp}(a/{\mathbb Z})$
, and say that
$ a$
is a realisation of
$ u$
, or that
$a$
generates
$u$
, to mean that
$u=\{A\subseteq {\mathbb Z}: a\in {}^{\ast } A\}$
. In other words, we identify ultrafilters in
$\beta {\mathbb Z}$
with
$1$
-types over
${\mathbb Z}$
in the language mentioned above.
In this setting, every type over
${\mathbb Z}$
is definable, and the product
$\otimes $
of ultrafilters coincides with the product
$\otimes $
of definable types, provided compatible conventions are adopted. Specifically, we have
$A\in u\otimes v\Leftrightarrow \{x: \{y: (x,y)\in A\}\in v\}\in u$
. In terms of realisations, this means that the order in which we resolve tensor products is reversed with respect to the majority of model-theoretic literature; namely, in this paper
$(a,b)\models u\otimes v$
iff
$a\models u$
and
$b\models v\mid {\mathbb Z} a$
.Footnote
1
In this case, we call
$(a,b)$
a tensor pair. Tensor pairs in
${}^{\ast }\mathbb {N}$
have been characterised by Puritz in [Reference Puritz, Luxemburg and Robinson9, Theorem 3.4]; we recall here the extension of Puritz’ characterisation to
${}^{\ast }\mathbb {Z}$
, and we refer to [Reference Di Nasso, Loeb and Wolff2, Section 11.5] or [Reference Luperi Baglini5] for a proof of this fact.
Fact 2.1. An ordered pair
$(a,b)\in {}^{\ast }\mathbb {Z}^{2}$
is a tensor pair if and only if for every
$f\colon \mathbb {Z}\to \mathbb {Z}$
either
${}^{\ast }f(b)\in {\mathbb {Z}}$
or
$\lvert a \rvert \leq \lvert {}^{\ast }f(b) \rvert $
.
The iterated hyper-extensions framework of nonstandard analysis allows for an even simpler characterisation of tensor products and related notions: if
$a,b\in {}^{\ast }\mathbb {Z}$
are such that
$a\models u$
and
$b\models v$
, then
$\left (a,{}^{\ast }b\right )\models u\otimes v$
. As a trivial consequence, in the same hypotheses we have that
$a+{}^{\ast }b\models u\oplus v$
and
$a\cdot {}^{\ast }b\models u\odot v$
. A detailed study of many properties and characterisations of tensor
$ k$
-tuples in this iterated nonstandard context can be found in [Reference Luperi Baglini5].
Let us recall (some equivalent forms of) the definitions of divisibility and congruence of ultrafilters. We will frequently use that, when dealing with generators of ultrafilters, some existential quantifiers may be replaced by universal ones. For example,
$(\exists a\models u)\; (\exists b\models v)\; a\mid b$
if and only if
$(\forall a\models u)\; (\exists b\models v)\; a\mid b$
, if and only if
$(\forall b\models v)\; (\exists a\models u)\; a\mid b$
. This follows from saturation of
${}^{\ast }{\mathbb Z}$
(see [Reference Luperi Baglini5, Corollary 5.13]). By this, and [Reference Šobot11, Proposition 3.2 and Theorem 4.5], we may take as definitions of
$\mathrel {\tilde \mid }$
and
$\equiv _w$
the ones below.Footnote
2
Similarly, our definition of
$\mathrel {\equiv ^{\mathrm {s}}_{w}}$
is not the original one, but it is equivalent to it by [Reference Šobot11, Lemma 6.5].
Definition 2.2. Let
$u,v,w\in \beta {\mathbb Z}$
, with
$w\ne 0$
.
-
(1) We write
$u\mathrel {\tilde \mid } v$
iff there are
$a\models u$
and
$b\models v$
such that
$a\mid b$
. -
(2) We write
$u\equiv _w v$
iff there are
$d\models w$
and
$(a,b)\models u\otimes v$
such that
$d\mid a-b$
. -
(3) We write
$u\mathrel {\mid ^{\mathrm {s}}} v$
iff there is
$(a,b)\models u\otimes v$
such that
$a\mid b$
. -
(4) We write
$u\mathrel {\equiv ^{\mathrm {s}}_{w}} v$
iff there is
$(d,a,b)\models w\otimes u\otimes v$
such that
$d\mid a-b$
.
We stress that the existential quantifier in the definition of
$\mathrel {\mid ^{\mathrm {s}}}$
may be replaced with a universal one: the property being checked is true of some realisation of the tensor product if and only if it is true of every realisation of the tensor product. The same holds for
$\mathrel {\equiv ^{\mathrm {s}}_{w}}$
, but not for
$\mathrel {\tilde \mid }$
, nor for
$\equiv _{w}$
: in the latter two cases, we can replace one (any) existential quantifier with an universal one, provided the universal quantifier is the leftmost one, as above, but not both simultaneously.
Remark 2.3. The following properties hold.
-
(1) The relation
$\mathrel {\tilde \mid }$
is a preorder. -
(2) The relation
$\mathrel {\mid ^{\mathrm {s}}}$
is transitive, but not reflexive (see later, or [Reference Šobot11, Lemma 6.4]). -
(3) We have
$u\equiv _w v$
if and only if
$w\mathrel {\tilde \mid } u\ominus v$
. -
(4) We have
$u\mathrel {\equiv ^{\mathrm {s}}_{w}} v$
if and only if
$w\mathrel {\mid ^{\mathrm {s}}} u\ominus v$
. -
(5) If
$w=n\ne 0$
is principal, then both
$\equiv _w$
and
$\mathrel {\equiv ^{\mathrm {s}}_{w}}$
coincide with the usual congruence relation modulo
$ n$
.
From Definition 2.2, it is easy to obtain nonstandard characterisations; for example,
$u\mathrel {\equiv ^{\mathrm {s}}_{w}} v$
if and only if whenever
$d,a,b\in {}^{\ast }\mathbb {Z}$
are such that
$d\models w, a\models u, b\models v$
, then
$d\mid {}^{\ast }a-{}^{\ast \ast }b$
. Below, we provide some further equivalent definitions of the divisibility relations.Footnote
3
Denote by
$\mathcal U$
the family of all
$\mid $
-upward closed subsets of
${\mathbb Z}$
.
Remark 2.4. For every
$u,v\in \beta {\mathbb Z}$
, the following hold.
-
(1) We have
$u\mathrel {\tilde \mid } v$
if and only if
$u\cap \mathcal U\subseteq v$
. -
(2) We have
$u\mathrel {\mid ^{\mathrm {s}}} v$
if and only if
$\{n\in {\mathbb Z}: n {\mathbb Z}\in v\}\in u$
.
Fact 2.5 [Reference Šobot11, Lemma 5.6 and Theorem 5.7].
For every
$w\in \beta {\mathbb Z} \setminus \{0\}$
, the relation
$\mathrel {\equiv ^{\mathrm {s}}_{w}}$
is an equivalence relation compatible with
$\oplus $
and
$\odot $
.
An important role in our analysis of these notions will be played by those ultrafilters which are maximal with respect to divisibility amongst nonzero ultrafilters.
Definition 2.6. We denote by
$\mathrm {MAX}$
the set of ultrafilters that are
$\mathrel {\tilde \mid }$
-divisible by all elements of
$\beta \mathbb {Z}\setminus \{0\}$
.
The following characterisation may be proven by taking suitable tensor products (see also [Reference Šobot11, Lemma 5.8(a)]).
Fact 2.7. The following are equivalent for
$w\in \beta {\mathbb Z} \setminus \{0\}$
.
-
(1) For every
$u\in \beta {\mathbb Z} \setminus \{0\}$
we have
$u\mathrel {\tilde \mid } w$
(that is,
$w\in \mathrm {MAX}$
). -
(2) For every
$u\in \beta {\mathbb Z} \setminus \{0\}$
we have
$u\mathrel {\mid ^{\mathrm {s}}} w$
. -
(3) For every
$n\in \mathbb N$
we have
$n\mathrel {\tilde \mid } w$
(that is,
$w\equiv _n 0$
, or equivalently
$n {\mathbb Z}\in w$
).
In the case of
$\beta \mathbb N$
, the following is [Reference Šobot12, Lemma 4.3]. Its version for
$\beta {\mathbb Z}$
is proven in the same way. Recall that
$K(\beta {\mathbb Z}, \odot )$
denotes the smallest bilateral ideal of the semigroup
$(\beta {\mathbb Z}, \odot )$
, and
$\overline {K(\beta {\mathbb Z}, \odot )}$
denotes its closure.
Fact 2.8. The set
$\mathrm {MAX}$
is topologically closed in
$\beta \mathbb {Z}$
. Moreover, it is a
$\odot $
-bilateral ideal and it is closed under
$\oplus $
. In particular,
$\overline {K(\beta \mathbb {Z}, \odot )}\subseteq \mathrm {MAX}$
.
Throughout, an important role will be played by the profinite completion
$\varprojlim {\mathbb Z}/n{\mathbb Z}$
of
$({\mathbb Z}, +)$
, which may be thought of as the additive group of consistent choices of remainder classes modulo each
$n\in \mathbb N$
, and is usually denoted by
$\hat {{\mathbb Z}}$
. Explicitly, we may identify an element of
$\hat {{\mathbb Z}}$
with a sequence
$(a_n)_{n\in \mathbb N}$
such that
$a_n\in \{0,\ldots , n-1\}$
and, if
$n\mid m$
, then
$a_m\equiv _n a_n$
, with pointwise addition modulo
$ n$
. There is an isomorphism
$(\hat {{\mathbb Z}},+)\cong \prod _{p\in \mathbb P} ({\mathbb Z}_p,+)$
, where
${\mathbb Z}_p$
denotes the
$ p$
-adic integers. Again up to isomorphism, we may view
$\hat {{\mathbb Z}}$
as the quotient of
$({}^{\ast } {\mathbb Z},+)$
by the equivalence relation that identifies
$ a$
and
$ b$
whenever, for every
$n\in \mathbb N$
, we have
$a\equiv _n b$
.
It is well-known that
$\hat {{\mathbb Z}}$
is a profinite group, that is, a topological group which is a Stone space, when equipped with the group topology where a basis of neighbourhoods of the identity is given by the clopen subgroups
$n \hat {{\mathbb Z}}$
. In other words, the basic (cl)open sets are given by fixing finitely many remainder classes. The isomorphism
$\hat {{\mathbb Z}}\cong \prod _{p\in \mathbb P} {\mathbb Z}_p$
is in fact an isomorphism of topological groups, that can be used to obtain a nice characterisation of the closed subgroups of
$(\hat {{\mathbb Z}}, +)$
. This may be proven directly, but it also follows from, e.g., Theorem 1.2.5 in [Reference Wilson14], to which we also refer the reader interested in an introduction to profinite groups. Below, we adopt the convention that, if
$\alpha $
is an infinite ordinal, then
$p^\alpha {\mathbb Z}_p=\{0\}$
.
Fact 2.9. View
$\hat {{\mathbb Z}}$
as
$\prod _{p\in \mathbb P} {\mathbb Z}_p$
. Then, the closed subgroups of
$(\hat {{\mathbb Z}}, +)$
are precisely those of the form
$\prod _{p\in \mathbb P} p^{\varphi (p)}{\mathbb Z}_p$
, where
$\varphi \colon \mathbb P\to \omega +1$
.
In particular, each closed subgroup may be written as
$\{x\in \hat {{\mathbb Z}}: \forall n\in D\; n \mid x\}$
, where D is a
$\mid $
-downward-closed subset of
${\mathbb Z}$
of the form
$\bigcap _{p\in \mathbb P}\left (p^{\varphi (p)+1}{\mathbb Z}\right )^{\mathsf {c}}$
.
3 Congruences that are not equivalences
We begin this section by proving that
$\equiv _w$
is not always an equivalence relation, thereby answering negatively a question of Šobot.
Example 3.1. Let
$w\in \beta {\mathbb Z}\setminus {\mathbb Z}$
be such that, for every
$n\in \mathbb N$
, we have
$w\equiv _n 1$
. Then
$\equiv _w$
is not transitive.
Proof For every
$w$
we have
$w\mathrel {\tilde \mid } (-w)$
, hence
$0\equiv _w w$
. On the other hand, for every
$w$
as above,
$w\ominus 1\in \mathrm {MAX}$
, so
$w\equiv _w 1$
, and by transitivity
$0\equiv _w 1$
, contradicting that
$w$
is non-principal.
Since any
$w$
of the form
$u\oplus 1$
, with
$u\in \mathrm {MAX}$
nonzero, satisfies the assumptions of Example 3.1, this settles [Reference Šobot11, Question 7.1]. In the rest of this section, we study in more detail how
$\equiv _w$
may fail to be an equivalence relation. We easily observe that reflexivity is always guaranteed.
Proposition 3.2. For all
$w\in \beta {\mathbb Z}\setminus \{0\}$
, the relation
$\equiv _w$
is reflexive.
Proof Given any
$u\in \beta \mathbb {Z}$
, observe that
$(a, a')\models u\otimes u$
implies that, for every
$n\in \mathbb {N}$
, we have
$a\equiv _n a'$
, hence that
$n\mathbb {Z}\in u\ominus u$
. By Fact 2.7
$u\ominus u\in \mathrm {MAX}$
, and we conclude by Remark 2.3.
As observed in Example 3.1, transitivity of
$\equiv _{w}$
is not guaranteed in general. Remarkably, failure of transitivity is the only obstruction to
$\equiv _w$
being an equivalence relation.
Theorem 3.3. For every
$w\in \beta {\mathbb Z}\setminus \{0\}$
, the relation
$\equiv _w$
is an equivalence relation if and only if it is transitive.
Proof By Proposition 3.2 we only need to show that if
$\equiv _w$
is transitive, then it is symmetric. Assume symmetry fails, as witnessed by
$u,v$
such that
$u\equiv _w v$
but
$v\not \equiv _w u$
. Let
and
. By construction
$t\equiv _w 0$
and
$t'\not \equiv _w 0$
. On the other hand
$t'\ominus t$
is easily checked to be in
$\mathrm {MAX}$
, hence
$t'\equiv _w t$
. It follows that
$t'\equiv _w t\equiv _w 0$
, but
$t'\not \equiv _w 0$
, so transitivity fails.
Symmetry of
$\equiv _{w}$
can also fail for reasons that have little to do with transitivity.Footnote
4
We will prove this by using upper Banach density, denoted by
$\operatorname {BD}$
. For definitions and basic properties around densities, see, e.g., [Reference Moreira, Richter and Robertson6] and the references therein. Specifically, we will use the following consequence of [Reference Bergelson1, Theorem 2.1] (see [Reference Moreira, Richter and Robertson6, Corollary 2.4] for more details).
Fact 3.4. Suppose that
$\{B_n\}_{n\in \mathbb {N}}$
is a family of subsets of
$\mathbb {Z}$
, that
$\Phi $
is a sequence of intervals of increasing length, and denote by
$d_\Phi $
the associated density. If there is
$\varepsilon>0$
such that, for every
$n\in \mathbb {N}$
, the density
$d_\Phi (B_n)$
exists and is larger than
$\varepsilon $
, then there is an infinite
$X\subseteq \mathbb {N}$
such that the family
$\{B_x: x\in X\}$
can be extended to a nonprincipal ultrafilter.
Theorem 3.5. Let
$A\subseteq \mathbb {Z}$
be such that
$\mathrm {BD}(A)>0$
and
$A^{\mathsf {c}}$
is thick. Then there are
$u, v\in \beta \mathbb {Z}\setminus \mathbb {Z}$
such that
$A\in u\oplus v$
and
$A^{\mathsf {c}}\in v\oplus u$
.
Proof Recall that a subset of
${\mathbb Z}$
is thick iff it contains arbitrarily long intervals. Hence, by assumption we can find, for every
$n\in \mathbb N$
, an interval
$J_n=[a_n, b_n]$
such that
$\lvert {J_n} \rvert =2n+1$
and
$J_n\subset A^{\mathsf {c}}$
. Denote by
$\Phi $
the sequence of intervals along which
$\mathrm {BD}(A)=d_\Phi (A)$
and observe that, for every
$m\in \mathbb {Z}$
, we have
$d_\Phi (A-m)=d_\Phi (A)$
.
Set
and apply Fact 3.4 with
, finding
$Y\subseteq \{c_n\}_{n\in \mathbb {N}}\subseteq A^{\mathsf {c}}$
such that
$\{A-y: y\in Y\}$
is contained in a nonprincipal ultrafilter
$ v$
.
Fix any nonprincipal ultrafilter
$ u$
containing Y. By construction,
$A\in u\oplus v$
, and we are left to show that
$A^{\mathsf {c}} \in v \oplus u$
. Because
$Y\subseteq \{c_n\}_{n\in \mathbb {N}}$
, for every
$a\in {}^{\ast }Y\setminus Y$
, hence in particular for every
$a\models u$
, and for every
$n\in \mathbb {Z}$
we have
$n+a\in {}^{\ast }A^{\mathsf {c}}$
. Therefore, if
$(b, a)\models v\otimes u$
, we have
$b+a\in {}^{\ast }A^{\mathsf {c}}$
, concluding the proof.
Corollary 3.6. There is
$w\in \beta {\mathbb Z}\setminus \{0\}$
such that
$\equiv _w$
is not symmetric.
Proof It is well-known that the set of squarefree integers has positive density, see, e.g., [Reference Jakimczuk4] for a short proof. Moreover, an easy application of the Chinese Remainder Theorem shows that its complement is thick: if
$p_k$
is the
$ k$
th prime, it suffices to find an integer
$ n$
such that
$n\equiv _{p_k^2}-k$
for sufficiently many
$ k$
. By Theorem 3.5, and the fact that squarefree integers form a symmetric subset of
${\mathbb Z}$
, there exist
$u, v\in \beta \mathbb {Z}\setminus \mathbb {Z}$
such that
$u\ominus v$
is squarefree (that is, it contains the set of squarefree integers; equivalently, its realisations are squarefree) and
$v\ominus u$
is not. Since
$v\ominus u$
is not squarefree, it is divided by some square
$w>1$
, which cannot divide the squarefree
$u\ominus v$
.
Remark 3.7. The proof of Corollary 3.6 also works if we fix an arbitrary
$\alpha \colon \mathbb {P}\to \omega +1$
such that
$\alpha (p)>1$
for every
$p\in \mathbb {P}$
and (recalling that we convene
$p^{\omega }\mathbb {Z}=\{0\}$
), replace the squarefree integers by
, the squarefree case corresponding to
$\alpha $
being constantly
$2$
. The set A has positive density because it contains the squarefree integers, and its complement is again proven to be thick by using the Chinese Remainder Theorem.
4 Self-divisible ultrafilters
In the previous section, we have seen examples of ultrafilters
$w$
such that
$\equiv _{w}$
is not an equivalence relation, namely, all those in
$\mathrm {MAX}\oplus 1$
except
$1$
. On the other hand, there are ultrafilters
$w$
such that
$\equiv _w$
is an equivalence relation, for instance, all principal ones.Footnote
5
It is natural to look for a characterisation of when this happens; in this section, we provide a complete solution to this problem.
Definition 4.1. Let
$w\in \beta \mathbb {Z}$
.
-
(1) We denote the set of integers dividing
$w$
by
. -
(2) We call
$w\in \beta {\mathbb Z}\setminus \{0\}$
self-divisible iff
$D(w)\in w$
.
Remark 4.2. By Remark 2.4,
$w\mathrel {\mid ^{\mathrm {s}}} u$
if and only if
$D(u)\in w$
if and only if for some (equivalently, every)
$(a,b)\models w\otimes u$
we have
$a\mid b$
. In particular,
$w$
is self-divisible if and only if
$w\mathrel {\mid ^{\mathrm {s}}} w$
. In nonstandard terms, this means that, whenever
$a\in {}^{\ast } {\mathbb Z}$
generates
$w$
, we have
$a\mid {}^{\ast } a$
.
As anticipated in the introduction, the relations
$\equiv _w$
and
$\mathrel {\equiv ^{\mathrm {s}}_{w}}$
have complementary shortcomings: the former is not, in general, an equivalence relation; as for the latter, there are ultrafilters
$w$
such that
$w\mathrel {\not \equiv ^{\mathrm {s}}_{w}}0$
, equivalently, such that
, for example, any
$w$
which is divided by no
$n>1$
. By definition, the self-divisible ultrafilters are those such that
$\mathrel {\equiv ^{\mathrm {s}}_{w}}$
is well-behaved in this respect. Perhaps unexpectedly, we will prove below that the self-divisible
$w$
are also precisely those for which the weak congruence
$\equiv _w$
is well-behaved, that is, is an equivalence relation.
We will do this via a small detour in the realm of profinite integers
$\hat {{\mathbb Z}}$
. This is no coincidence, since we will later show in Remark 7.1 that
$(\beta {\mathbb Z}, \oplus )/\mathord {\mathrel {\equiv ^{\mathrm {s}}_{w}}}$
is isomorphic to a quotient of
$\hat {{\mathbb Z}}$
. The connection between
$\beta {\mathbb Z}$
and
$\hat {{\mathbb Z}}$
arises naturally from the fact that every
$u\in \beta {\mathbb Z}$
induces a consistent choice of remainder classes modulo the standard integers, that is, an element of
$\hat {{\mathbb Z}}$
. Let us give a name to the corresponding function.
Definition 4.3. Define
$\pi \colon \beta {\mathbb Z} \to \hat {{\mathbb Z}}$
as the map sending each ultrafilter to the sequence of its remainder classes.
Our detour will lead us to talk about the following sets.
Definition 4.4. We denote by
the set of ultrafilters divisible by
$w$
.
Remark 4.5. By Remark 2.4 the set
$Z_w$
is a closed subset of
$\beta {\mathbb Z}$
, corresponding to the filter of
$\mid $
-upward-closed elements of
$w$
.
The remark above has a converse. Since we will never use it, we leave the (standard) proof to the reader.
Remark 4.6. If
$\mathcal F\subseteq \mathcal U$
is a family of
$\mid $
-upward-closed subsets of
${\mathbb Z}$
, then there is
$w\in \beta {\mathbb Z}$
such that
$\mathcal F=w\cap \mathcal U$
if and only if
$\mathcal F$
is a prime filter on the distributive lattice
$\mathcal U$
.
Lemma 4.7. The following statements hold.
-
(1) The map
$\pi $
is continuous, surjective, and a homomorphism of semigroups
$(\beta {\mathbb Z}, \oplus )\to (\hat {{\mathbb Z}}, +)$
. -
(2) The quotient topology induced by
$\pi $
coincides with the usual topology of
$\hat {{\mathbb Z}}$
. -
(3) If
$Z_w$
is closed under
$\oplus $
, then
$\pi (Z_w)$
is a closed subgroup of
$(\hat {{\mathbb Z}}, +)$
.
Proof The first point is an easy exercise, and the second one follows from the facts that every continuous surjective map from a compact space to a Hausdorff one is closed, and that every continuous, surjective, closed map is automatically a quotient map. As for the last point, (for every
$w$
) the set
$Z_w$
contains
$0$
and is closed under
$-$
. Grouphood of
$\pi (Z_w)$
then follows easily from the assumption, and we conclude by observing that both spaces are compact Hausdorff.
Lemma 4.8. For every
$w\in \beta {\mathbb Z} \setminus \{0\}$
and every
$u,u^{\prime }\in \beta \mathbb {Z}$
, if
$\pi (u)=\pi (u')$
, then
$u\mathrel {\equiv ^{\mathrm {s}}_{w}} u'$
.
Proof If
$\pi (u)=\pi (u')$
, then
$u\ominus u'\in \mathrm {MAX}$
, and we conclude by Fact 2.7 and Remark 2.3.
Lemma 4.9. If
$\equiv _w$
is an equivalence relation, then it is a congruence with respect to
$\oplus $
.
Proof Assume that
$u\equiv _w v$
, and let us show that for all
$t\in \beta \mathbb {Z}$
we have
$u\oplus t\equiv _w v\oplus t$
. By definition,
$u\ominus v\equiv _w 0$
, and it is easy to see that
$\pi (u\ominus v)=\pi (u\oplus t \ominus v \ominus t)$
. By Lemma 4.8 we have
$u\ominus v\mathrel {\equiv ^{\mathrm {s}}_{w}} u\oplus t \ominus v \ominus t$
. Whenever
$\equiv _w$
is an equivalence relation, it is automatically a coarser one than
$\mathrel {\equiv ^{\mathrm {s}}_{w}}$
by definition, so
$u\oplus t \ominus v \ominus t\equiv _w 0$
, hence
$u\oplus t \equiv _w v\oplus t$
. The proof that
$t\oplus u \equiv _w t\oplus v$
is analogous.
We are now ready to prove the first part of our main result. Later, in Theorem 7.7, we will see several more properties equivalent to self-divisibility.
Theorem 4.10. For every
$w\in \beta {\mathbb Z} \setminus \{0\}$
, the following are equivalent.
-
(1) The ultrafilter
$w$
is self-divisible. -
(2) The relations
$\equiv _w$
and
$\mathrel {\equiv ^{\mathrm {s}}_{w}}$
coincide. -
(3) The relation
$\equiv _w$
is an equivalence relation. -
(4) For every
$ u$
, we have
$w\mathrel {\tilde \mid } u$
if and only if
$D(w)\subseteq D(u)$
. -
(5) For every
$a,b\models w$
there is
$c\models w$
such that
$c\mid \gcd (a,b)$
.
Proof In order to prove (1)
$\Rightarrow $
(2), we need to show that if
$u\equiv _w v$
then
$u\mathrel {\equiv ^{\mathrm {s}}_{w}} v$
, since the converse is always true. Let
$d\models w$
and
$(a,b)\models u\otimes v$
be such that
$d\mid a-b$
. Let
and find, using saturation, some
$d'$
such that
$(d', (d,a,b))\models w \otimes t$
. In particular,
$(d', (a,b))$
is a tensor pair, hence by associativity of
$\otimes $
, we have
$(d',a,b)\models w\otimes u\otimes v$
, therefore
$u\mathrel {\equiv ^{\mathrm {s}}_{w}} v$
if and only if
$d'\mid a-b$
. But
$(d', d)\models w\otimes w$
, hence
$d'\mid d$
by assumption and Remark 4.2.
The implication (2)
$\Rightarrow $
(3) is obvious because
$\mathrel {\equiv ^{\mathrm {s}}_{w}}$
is an equivalence relation by Fact 2.5.
To prove (3)
$\Rightarrow $
(4), assume that
$\equiv _w$
is an equivalence relation. It is easy to see, for instance, by using Remark 2.4, that
$w\mathrel {\tilde \mid } u\Rightarrow D(w)\subseteq D(u)$
always holds, so let us assume that
$D(w)\subseteq D(u)$
. We need to prove that
$w\mathrel {\tilde \mid } u$
or, in other words, that
$u\in Z_w$
. We begin by observing that, for every
$ v$
,
$v'$
, if
$\pi (v)=\pi (v')$
, then
$v\mathrel {\equiv ^{\mathrm {s}}_{w}} v'$
by Lemma 4.8, hence, since we are assuming
$\equiv _w$
is an equivalence relation,
$v\in Z_w\Leftrightarrow v'\in Z_w$
, so it suffices to show that
$\pi (u)\in \pi (Z_w)$
. By Lemma 4.9,
$\equiv _w$
is a congruence with respect to
$\oplus $
, thus
$Z_w$
is closed under
$\oplus $
. By Lemma 4.7,
$\pi (Z_w)$
is a closed subgroup of
$(\hat {{\mathbb Z}}, +)$
, therefore, by Fact 2.9, there is a
$\mid $
-downward closed
$D\subseteq {\mathbb Z}$
such that
$\pi (Z_w)=\{x\in \hat {{\mathbb Z}}: \forall n\in D\; n \mid x\}$
. In other words, for every
$ v$
, we have
$\pi (v)\in \pi (Z_w)\Leftrightarrow D\subseteq D(v)$
. Trivially,
$\pi (w)\in \pi (Z_w)$
, hence
$D\subseteq D(w)$
, and we conclude by using the assumption that
$D(w)\subseteq D(u)$
.
For (4)
$\Rightarrow $
(5), observe that if
$n\mathrel {\tilde \mid } w$
then automatically
$n\mid \gcd (a,b)$
. Hence
$D(w)\subseteq D(\textrm {tp}(\gcd (a,b)/{\mathbb Z}))$
, and the conclusion follows.
To prove (5)
$\Rightarrow $
(1), let
$(a,b)\models w\otimes w$
. By assumption, there is
$c\models w$
such that
$c\mid \gcd (a,b)$
. This implies that
$\lvert c \rvert \le \lvert a \rvert $
, hence by Fact 2.1 we have
$(c,b)\models w\otimes w$
, witnessing self-divisibility.
5 Examples
In this section we look at some examples and non-examples of self-divisible ultrafilters and see how this notion interacts with other fundamental classes of ultrafilters, such as the idempotent or the minimal elements of the semigroups
$(\beta \mathbb N, \oplus )$
and
$(\beta \mathbb N, \odot )$
. We also define a special kind of self-divisible ultrafilters, the division-linear ones, and look at the relation between the shape of
$D(w)$
and self-divisibility of
$w$
.
Example 5.1.
-
(1) Clearly, every nonzero principal ultrafilter is self-divisible.
-
(2) Every ultrafilter in
$\mathrm {MAX}$
is easily checked to be self-divisible. -
(3) Every ultrafilter of the form
$\textrm {tp}(a/{\mathbb Z})$
, where
$a> {\mathbb Z}$
is prime, is not self-divisible. -
(4) If
$p_1,\ldots , p_n\in \mathbb P$
, and
$a_1,\ldots , a_n, b\in {}^{\ast } \mathbb N$
, then
$\textrm {tp}(p_1^{a_1}\cdot \ldots \cdot p_n^{a_n}\cdot b/{\mathbb Z})$
is self-divisible if and only if
$\textrm {tp}(b/{\mathbb Z})$
is. This can be easily seen by using point (5) of Theorem 4.10. -
(5) In particular, every ultrafilter of the form
$\textrm {tp}(p_1^{a_1}\cdot \ldots \cdot p_n^{a_n}/{\mathbb Z})$
is self-divisible. -
(6) Self-divisible ultrafilters form a semigroup with respect to
$\odot $
. -
(7) By (2) above and Fact 2.8, all
$\odot $
-minimal ultrafilters are self-divisible. -
(8) By (2) above, all
$\oplus $
-idempotent ultrafilters are self-divisible. -
(9) If
$u\ne 0$
is a minimal
$\oplus $
-idempotent, then
$u\oplus 1$
is
$\oplus $
-minimal, but not self-divisible, since
$D(u\oplus 1)=\{1, -1\}$
.
We will see in Example 5.4 that, in (9) above, the reverse inclusion does not hold either. In fact,
$\oplus $
-minimality is not even implied by the following strengthening of self-divisibility.
Definition 5.2. An ultrafilter
$w\in \beta \mathbb {Z}$
is division-linear iff
$w\ne 0$
and for every pair
$d,d'$
of realisations of
$w$
we have that
$d\mid d'$
if and only if
$\lvert {d} \rvert \leq \lvert {d'} \rvert $
.
We already said that every nonzero principal ultrafilter is self-divisible; in fact, every such ultrafilter is division-linear. We now look at another example, then characterise division-linearity. For a similar characterisation of self-divisibility, see Theorem 7.7.
Example 5.4. Every ultrafilter
$ u$
containing the set
of factorials is division-linear. Because F is not a multiplicative IP set,
$ u$
is not
$\odot $
-idempotent and, because F is not piecewise syndetic,
$ u$
is not in
$\overline {K(\beta \mathbb {Z}, \oplus )}$
.
Proposition 5.5. A nonzero
$w\in \beta {\mathbb Z}$
is division-linear if and only if it contains a set linearly preordered by divisibility.
Proof If
$A\in w$
is linearly preordered by divisibility, then the conclusion follows by observing that whenever
$d,d'\models w$
then
$d,d'\in {}^{\ast }A$
. Conversely, if we think of
$w$
as a type, then
$w$
is division-linear, by definition, if and only if
$w(x)\cup w(y) \vdash (\lvert x \rvert \le \lvert y \rvert ) \to (x\mid y)$
. By compactness, there are
$A, B\in w$
such that
$(x\in {}^{\ast }A) \land (y\in {}^{\ast }B) \vdash (\lvert x \rvert \le \lvert y \rvert ) \to (x\mid y)$
. It follows that
$A\cap B$
is linearly preordered by divisibility.
Proposition 5.6. There is an ultrafilter in
$\mathrm {MAX}$
(in particular, a self-divisible ultrafilter) which is not division-linear.
Proof Let
$\mathcal L$
be the family of subsets of
${\mathbb Z}\setminus \{0\}$
which are linearly preordered by divisibility. By Proposition 5.5, it suffices to prove that the family
has the finite intersection property. But this is clear, because
$\{n {\mathbb Z} : n>1\}$
is closed under finite intersections and every
$n {\mathbb Z}$
contains an infinite
$\mid $
-antichain, hence cannot be contained in a finite union of elements of
$\mathcal L$
.
The reverse inclusion also fails:
Example 5.7. If
$ u$
is the type of a nonstandard power of
$2$
, then it is division-linear, hence self-divisible, but not in
$\mathrm {MAX}$
.
Example 5.8. Self-divisibility is not preserved upwards nor downwards by
$\mathrel {\tilde \mid }$
. For instance, if
$v\in \mathrm {MAX}$
, then for any non-self-divisible
$w$
we have both
$w\mathrel {\tilde \mid } v$
and
$w\mathrel {\mid ^{\mathrm {s}}} v$
(Fact 2.7). In the other direction, fix infinite
$a,b$
with
$ b$
prime and take
and
. It is easy to show that
$w(x)\otimes v(y)\vdash x\mid y$
, from which we deduce
$w\mathrel {\mid ^{\mathrm {s}}} v$
, and in particular
$w\mathrel {\tilde \mid } v$
. By Example 5.7
$w$
is division-linear, hence self-divisible, but
$ v$
is not.
We saw in point (7) of Example 5.1 and in Example 5.4 that
$\odot $
-minimal ultrafilters are self-divisible, and that division-linearity does not imply
$\odot $
-idempotency. Moreover, it is easily seen that every nonprincipal ultrafilter containing the set of primes is neither self-divisible, nor
$\odot $
-idempotent. Proposition 5.9 and Corollary 5.12 complete the picture.
Proposition 5.9. There exist
$\odot $
-idempotent non-self-divisible ultrafilters.
Proof Recall that an ultrafilter is
$\mathbb {N}$
-free iff it is not divisible by any
$n>1$
(see [Reference Šobot11] and the references therein). Denote the set of
$\mathbb {N}$
-free ultrafilters by
. Let us show that this set is closed under
$\odot $
. Indeed,
$w\in \mathrm {Free}$
if and only if for every
$a\models w$
and for every
$n>1$
we have
$n\nmid a$
. If
$(a, b)\models w\otimes v$
and
$n\mid a\cdot b$
, then every prime divisor of
$ n$
must divide either
$ a$
or
$ b$
, which is a contradiction because
$n>1$
. From this it follows easily that
$\mathrm {Free}\setminus \{1, -1\}$
is a closed subset of
$\beta \mathbb {Z}$
closed under
$\odot $
. It is therefore a compact right topological semigroup, and by Ellis’ Lemma it must contain a
$\odot $
-idempotent. To conclude, note that
$\mathrm {Free}$
does not contain any nonprincipal self-divisible ultrafilters, since every
$w\in \mathrm {Free}$
has
$D(w)=\{1,-1\}$
.
Lemma 5.10. If
$A\subseteq \mathbb N$
is linearly ordered by divisibility, then A contains no arithmetic progression of length
$3$
.
Proof A counterexample
$a,a+b, a+2b\in A$
should satisfy
$(a+b)\mid (a+b)+b$
, hence
$(a+b)\mid b$
, a contradiction.
The following fact is well-known, easy to prove, and a special case of the much more general [Reference Luperi Baglini5, Example 5.6].
Fact 5.11. The set of ultrafilters
$ u$
such that every element of
$ u$
contains an arithmetic progression of length
$3$
is a closed bilateral ideal of
$(\beta \mathbb N, \odot )$
.
Corollary 5.12. There is no nonzero, division-linear ultrafilter in
$\overline {K(\beta \mathbb {N}, \odot )}$
.
We thank the referee for catching a mistake in a previous version of the proof below.
Proof If
$u\in \beta \mathbb N$
is a counterexample, by Proposition 5.5 and the fact that we are working over
$\mathbb N$
, some
$A\in u$
is linearly ordered by divisibility, hence by Lemma 5.10 it contains no arithmetic progression of length
$3$
. This contradicts
$u\in \overline {K(\beta \mathbb {N}, \odot )}$
by Fact 5.11.
Self-divisible ultrafilters are not closed under
$\oplus $
: it suffices to sum any
$u\in \mathrm {MAX}$
with
$v=1$
. In fact, it is possible to construct a counterexample with both
$u,v$
nonprincipal.
Proposition 5.13. There are division-linear, nonprincipal ultrafilters
$u,v$
such that
$u\oplus v$
is not self-divisible.
Proof Let
$f\colon \mathbb N\to \mathbb P$
be the increasing enumeration of all primes, and fix
$a\in {}^{\ast }{\mathbb Z}$
such that
$a> {\mathbb Z}$
. Let
and
. It is easy to see that
$ u$
,
$ v$
are division-linear. Because every
$p\in \mathbb P$
divides precisely one between
$ u$
and
$ v$
, we have
$D(u\oplus v)=\{1,-1\}$
. Since
$u\oplus v$
is nonprincipal, it cannot be self-divisible.
Point (4) of Theorem 4.10 might suggest that self-divisibility of
$w$
could be deduced just by looking at
$D(w)$
. Rather surprisingly, this is false, except in some trivial cases. To prove this, it will be convenient to replace
$D(w)$
by the following function on the set
$\mathbb {P}$
of prime natural numbers. The reader familiar with algebra may recognise that such functions are exactly the same as supernatural numbers in the sense of Steinitz.
Definition 5.14. Given
$u\in \beta \mathbb {Z}$
, we define
$\varphi _u\colon \mathbb {P}\to \omega +1$
as the function sending
$p$
to
$\max \{k\in \omega : p^k\mathbb {Z}\in u\}$
if this exists, and to
$\omega $
otherwise.
Remark 5.15.
-
(1) By definition, for every
$u\in \beta \mathbb {Z}$
, the set
$D(u)$
determines
$\varphi _u$
, and conversely. More explicitly, recalling our convention that
$p^{\omega +1}\mathbb {Z}=\{0\}$
, we have
$$\begin{align*}D(u)=\bigcap_{p\in \mathbb{P}}(p^{\varphi_u(p)+1}\mathbb{Z})^{\mathsf{c}}. \end{align*}$$
-
(2) The set
(†)is
$$ \begin{align} D(u)^{\mathsf{c}}=\{k: k\mathbb{Z} \not\in u\}=\bigcup_{p\in \mathbb{P}}p^{\varphi_u(p)+1}\mathbb{Z} \end{align} $$
$\mid $
-upward closed.
Definition 5.16. Let
$\varphi \colon \mathbb {P}\to \omega +1.$
We say that
$\varphi $
is finite iff
$\varphi ^{-1}(\{\omega \})=\emptyset $
and
$\varphi ^{-1}(\mathbb N)$
is finite.
We say that
$\varphi $
is cofinite iff
$\varphi ^{-1}(\{0\})\cup \varphi ^{-1}(\mathbb N)$
is finite, that is,
$\varphi ^{-1}(\{\omega \})$
is cofinite.
Intuitively,
$\varphi _u$
is finite whenever
$ u$
is only divisible by finitely many integers, and
$\varphi _u$
is cofinite whenever, for all but finitely many
$p\in \mathbb P$
, the ultrafilter
$ u$
is divisible by every power of
$p$
.
Proposition 5.17. Let
$\varphi \colon \mathbb {P}\to \omega +1$
.
-
(1) If
$\varphi $
is finite, then every
$w\in \beta \mathbb {Z}$
such that
$\varphi _w=\varphi $
is either principal or not self-divisible. -
(2) If
$\varphi $
is cofinite, then every
$w\in \beta \mathbb {Z}$
such that
$\varphi _w=\varphi $
is self-divisible. -
(3) In every other case, namely whenever
$\varphi $
is not finite neither cofinite, there exist
$u, v\in \beta \mathbb {Z}$
such that
$\varphi _u=\varphi _v=\varphi $
with
$ u$
self-divisible and
$ v$
not self-divisible.
Proof The first point is immediate from the fact that, if
$\varphi _w$
is finite, then
$D(w)$
is finite. As for the second point, if
$\varphi _w$
is cofinite then, by definition, for every
$p$
outside of a certain finite
$P_0\subseteq \mathbb P$
we have
$p^{\varphi (p)+1}\mathbb {Z}=\{0\}$
. In other words, the union in (†) is actually a finite union, namely,
$$\begin{align*}D(w)^{\mathsf{c}}=\bigcup_{p\in P_0}p^{\varphi(p)+1}\mathbb{Z}.\end{align*}$$
By definition of
$\varphi $
, this is a finite union of sets not in
$w$
, hence does not belong to
$w$
.
Towards the last point, define
Every ultrafilter
$w$
extending
$\mathcal F$
will have
$D(w)=(\bigcup _{p\in \mathbb {P}}p^{\varphi (p)+1}\mathbb {Z})^{\mathsf {c}}$
, so we need to prove that, if
$\varphi $
is not finite nor cofinite, then the families below have the finite intersection property
$$\begin{align*}{\mathcal F}\cup\Big\{\bigcup_{p\in \mathbb{P}}p^{\varphi(p)+1}\mathbb{Z}\Big\}; \qquad \mathcal F \cup\Big\{\Big(\bigcup_{p\in \mathbb{P}}p^{\varphi(p)+1}\mathbb{Z}\Big)^{\mathsf{c}}\Big\}. \end{align*}$$
If
$\mathcal I\subseteq \mathcal F$
is finite, then its intersection may be written as follows, for some
$p_1, \ldots , p_k\in \varphi ^{-1}(\mathbb N)$
, some
$q_1, \ldots , q_s\in \varphi ^{-1}(\{0\})$
, some
$r_1, \ldots , r_\ell \in \varphi ^{-1}(\{\omega \})$
, and some
$n_1, \ldots , n_\ell \in \mathbb {N}$
:
$$ \begin{align*} \bigcap \mathcal{I}&= p_1^{\varphi(p_1)}\mathbb{Z}\cap\cdots\cap p_k^{\varphi(p_k)}\mathbb{Z}\cap (p_1^{\varphi(p_1)+1}\mathbb{Z})^{\mathsf{c}}\cap\cdots\cap (p_k^{\varphi(p_k)+1}\mathbb{Z})^{\mathsf{c}}\\ &\cap(q_1\mathbb{Z})^{\mathsf{c}}\cap\cdots\cap(q_s\mathbb{Z})^{\mathsf{c}}\\ &\cap r_1^{n_1}\mathbb{Z}\cap\cdots\cap r_\ell^{n_\ell}\mathbb{Z}. \end{align*} $$
Define
and observe that it belongs to
$\bigcap \mathcal I$
. By definition, if
$\varphi ^{-1}(\mathbb N)$
is infinite, then automatically
$\varphi $
is neither finite nor cofinite. Infinity of
$\varphi ^{-1}(\mathbb N)$
gives us some
$p_\dagger \in \varphi ^{-1}(\mathbb N)\setminus \{p_1, \ldots , p_k\}$
, so we get that
$a\cdot p_\dagger ^{\varphi (p_\dagger )+1}\in \bigcap \mathcal {I}\cap \bigcup _{p\in \mathbb {P}}p^{\varphi (p)+1}\mathbb {Z}$
, and that
$a\cdot p_\dagger \in \bigcap \mathcal {I}\cap \left (\bigcup _{p\in \mathbb {P}}p^{\varphi (p)+1}\mathbb {Z}\right )^{\mathsf {c}}$
.
If instead
$\varphi ^{-1}(\mathbb N)$
is finite then, because we are assuming that
$\varphi $
is not cofinite, the set
$\varphi ^{-1}(\{0\})$
is infinite. Let
$ a$
be as above. If
$q\in \varphi ^{-1}(\{0\})\setminus \{q_1, \ldots , q_s\}$
, then
$a\cdot q\in \bigcap \mathcal {I}\cap \bigcup _{p\in \mathbb {P}}p^{\varphi (p)+1}\mathbb {Z}$
. Moreover,
$a\in \mathcal {I}\cap \left (\bigcup _{p\in \mathbb {P}}p^{\varphi (p)+1}\mathbb {Z}\right )^{\mathsf {c}}$
, and we are done.
6 A bit of topology
In this section we study the topological properties of the subspaces of self-divisible and of division-linear ultrafilters. We begin with an easy remark.
Remark 6.1. For every
$u\in \beta {\mathbb Z}$
we have the following.
-
(1) [Reference Šobot10, Lemma 1.3] The set
$\{w: w\mathrel {\tilde \mid } u\}$
is closed: it coincides with This follows from Remark 2.4 and the fact that A is
$$\begin{align*}\bigcap \{\overline{B}: B\in u, B \text { is } {\mid}\text{-downward closed}\}. \end{align*}$$
$\mid $
-upward closed if and only if
$A^{\mathsf {c}}$
is
$\mid $
-downward closed.
-
(2) The set
$\{w: w\mathrel {\mid ^{\mathrm {s}}} u\}$
is clopen: it coincides with
$\overline {D(u)}$
.
The set of division-linear ultrafilters contains all the nonzero principal ultrafilters, and it follows that its closure is
$\beta {\mathbb Z}\setminus \{0\}$
. Therefore, we look at the topological properties of self-divisible and division-linear ultrafilters in the subspace of nonprincipal ultrafilters.
Definition 6.2. Let
$\mathrm {SD}$
denote the set of self-divisible nonprincipal ultrafilters. Similarly, denote by
$\mathrm {DL}$
the set of division-linear nonprincipal ultrafilters. Let
$\overline {\mathrm {SD}}, \overline {\mathrm {DL}}$
be their topological closures in
$\beta \mathbb {Z}\setminus {\mathbb Z}$
.
Proposition 6.3. We have
$\overline {\mathrm {SD}} = \overline {\mathrm {DL}}$
= {
$w$
∈ βℤ: ∀A ∈
$w$
∃X
$\subseteq$
A (X is an infinite
$\mid$
-chain)}.
Proof Let Ψ(
$w$
):= ∀A ∈
$w$
∃X
$\subseteq$
A (X is an infinite
$\mid$
-chain). First of all, we show that
$\Psi (w)$
holds for every
$w\in \mathrm {SD}$
. For such a
$w$
, by definition
$D(w)\in w$
. Fix
$A\in w$
and let
$x_1\in A\cap D(w)\in w$
. Since
$x_1\in D(w)$
, we have
$x_1\mathbb {Z}\in w$
. If we take
$x_2\in x_1\mathbb {Z}\cap A\cap D(w)\in w$
such that
$\lvert {x_2} \rvert>\lvert {x_1} \rvert $
, then
$x_2\mathbb {Z}\in w$
, and by induction we obtain the desired infinite chain
$X=\{x_1\mid x_2\mid \ldots \}\subseteq A$
.
If
$\Psi (w)$
holds, and
$A\in w$
, then every nonprincipal
$ u$
containing an infinite linearly ordered
$X\subseteq A$
must be division-linear. Therefore, in every open neighbourhood of
$w$
we can find an element of
$\mathrm {DL}$
, hence
$\Psi (w)$
implies that
$w\in \overline {\mathrm {DL}}$
. Conversely, assume
$w\in \overline {\mathrm {DL}}$
. Then for every
$A\in w$
there exists
$u\in \mathrm {DL}$
such that
$A\in u$
. If
$X\in u$
witnesses division-linearity of
$u$
, then
$X\cap A\in u$
is a linearly ordered infinite subset of A, hence
$\Psi (w)$
holds.
We conclude by observing that
$\mathrm {DL}\subseteq \mathrm {SD}\subseteq \overline {\mathrm {DL}}$
implies
$\overline {\mathrm {SD}}=\overline {\mathrm {DL}}$
.
We call additive (multiplicative, respectively) Hindman ultrafilters those in the closure of the nonprincipal
$\oplus $
-idempotents (
$\odot $
-idempotents, respectively). Because
$\oplus $
-idempotents are in
$\mathrm {MAX}$
, which is topologically closed, every additive Hindman ultrafilter belongs to
$(\mathrm {MAX}\setminus \{0\})\subseteq \mathrm {SD}$
.
Corollary 6.4. Every multiplicative Hindman ultrafilter is in
$\overline {\mathrm {SD}}$
.
Proof An ultrafilter
$w$
is multiplicatively Hindman if and only if for every
$A\in w$
there exists an increasing sequence
$(x_i)_{i\in \omega }$
of integers such that
is a subset of A. Now, notice that
$\{x_0\cdot \ldots \cdot x_k: k\in \omega \}$
is a proper subset of
$\operatorname {FP}((x_i)_{i\in \omega })\subseteq A$
linearly ordered by divisibility. By Proposition 6.3 we conclude that
$w\in \overline {\mathrm {SD}}$
.
Combining Corollary 6.4 and Propositions 5.9 and 6.3, we obtain the following.
Corollary 6.5. The sets
$\mathrm {SD}$
and
$\mathrm {DL}$
are not closed in
$\beta \mathbb {Z} \setminus \mathbb {Z}$
.
7 A bit of (topological) algebra
We study the quotients
$\beta {\mathbb Z}/\mathord {\mathrel {\equiv ^{\mathrm {s}}_{w}}}$
and prove some additional characterisations of self-divisibility.
Diagram from Remark 7.1.
Recall that every relation
$\mathrel {\equiv ^{\mathrm {s}}_{w}}$
is a congruence with respect to
$\oplus $
(Fact 2.5).
Remark 7.1.
-
(1) Since, by Lemma 4.8, the
$\mathrel {\equiv ^{\mathrm {s}}_{w}}$
-class of
$u\in \beta {\mathbb Z}$
only depends on its image in
$\hat { {\mathbb Z}}$
, the quotient map
$\rho _w\colon \beta {\mathbb Z}\to \beta {\mathbb Z}/\mathord {\mathrel {\equiv ^{\mathrm {s}}_{w}}}$
factors through a well-defined map
$\sigma _w\colon \hat {\mathbb Z}\to \beta {\mathbb Z}/\mathord {\mathrel {\equiv ^{\mathrm {s}}_{w}}}$
, which sends
$\pi (u)$
to
$u/\mathord {\mathrel {\equiv ^{\mathrm {s}}_{w}}}$
. Note that
$\sigma _w$
is a homomorphism of groups. -
(2) Let
$w\in \beta {\mathbb Z}$
be such that
$w>1$
, and view
$\hat {{\mathbb Z}}$
as a subgroup of
$\prod _{n\ge 2} {\mathbb Z}/{n {\mathbb Z}}$
. Let
$\hat {{\mathbb Z}}/w$
be the image of
$\hat {{\mathbb Z}}$
under the projection from
$\prod _{n\ge 2} {\mathbb Z}/{n {\mathbb Z}}$
onto the ultraproduct
$\prod _w {\mathbb Z}/{n {\mathbb Z}}$
. The obvious map
$\beta {\mathbb Z}/\mathord {\mathrel {\equiv ^{\mathrm {s}}_{w}}}\to \hat {{\mathbb Z}}/w$
is (well-defined and) an isomorphism making the diagram in Figure 1 commute. By using commutativity of the diagram, together with the fact that
$\pi $
is a closed map, it is easy to check that
$\sigma _w$
is continuous with respect to the quotient topology on
$\beta {\mathbb Z}/\mathord {\mathrel {\equiv ^{\mathrm {s}}_{w}}}$
, and in fact induces the same quotient topology, that is,
$\sigma _w^{-1}(C)$
is closed if and only if C is closed (if and only if
$\rho _w^{-1}(C)$
is closed). -
(3) In particular, if
$w$
is self-divisible then, by Theorem 4.10 and Lemma 4.7, the sequences
$(k_n)_{n\ge 2}$
such that
$k_n=0$
for
$w$
-almost every
$ n$
form a closed subgroup of
$\hat {{\mathbb Z}}$
, namely,
$\pi (Z_w)$
, which then coincides with the kernel of the projection
$\hat { {\mathbb Z}}\to \hat {{\mathbb Z}}/w$
. By a standard fact about profinite groups (see, e.g., [Reference Wilson14, Theorem 1.2.5]),
$\hat {{\mathbb Z}}/w$
with the quotient topology induced by this projection is profinite. We will see in Theorem 7.7 that the converse is also true, namely, that
$\ker (\sigma _w)$
is closed if and only if
$w$
is self-divisible.
Corollary 7.2. The quotient
$\beta {\mathbb Z}/\mathord {\mathrel {\equiv ^{\mathrm {s}}_{w}}}$
may be identified with a subgroup of the ultraproduct
$\prod _w {\mathbb Z}/n {\mathbb Z}$
, which is isomorphic to a quotient (as abstract groups) of
$\hat {{\mathbb Z}}$
. If
$w$
is self-divisible, then it is isomorphic to
$\prod _{p \in \mathbb P} G_p$
, where
$G_p= {\mathbb Z}_p$
if
$\varphi _w(p)=\omega $
, and
$G_p= {\mathbb Z}/p^{\varphi _w(p)}{\mathbb Z}$
otherwise.
Proof This follows at once from Remark 7.1, Fact 2.9, and the fact that
$\hat {{\mathbb Z}}\cong \prod _{p\in \mathbb P} {\mathbb Z}_p$
.
Proposition 7.3. The map
$\sigma _w$
is injective if and only if it is an isomorphism, if and only if
$w\in \mathrm {MAX}\setminus \{0\}$
.
Proof If
$w=0$
the map
$\sigma _w$
is not defined, so let
$w\ne 0$
. Assume
$w\in \mathrm {MAX}$
, and observe that
$\pi ^{-1}(\{0\})=\mathrm {MAX}$
. So if
$\pi (v)\ne 0$
then
$v\notin \mathrm {MAX}$
, hence
by Fact 2.7. This shows that, if
$\pi (v)\ne 0$
, then
$\sigma _w(\pi (v))\ne 0$
, so
$\sigma _w$
is injective. Conversely, if
$w\notin \mathrm {MAX}$
, there must be
$n>1$
such that
$(n {\mathbb Z})^{\mathsf {c}}\in w$
. If
$n=p_0^{k_0}\cdot \ldots \cdot p_\ell ^{k_\ell }$
, then
$(n {\mathbb Z})^{\mathsf {c}}= (p_0^{k_0}{\mathbb Z})^{\mathsf {c}}\cup \cdots \cup (p_\ell ^{k_\ell }{\mathbb Z})^{\mathsf {c}}$
. Without loss of generality
$(p_0^{k_0}{\mathbb Z})^{\mathsf {c}}\in w$
. Take any
$ v$
congruent to
$p_0^{k_0-1}$
modulo every power of
$p_0$
and divided by every other prime power; in other words, take
$ v$
with
$\varphi _v(p_0)=k_0-1$
and
$\varphi _v(p')=\omega $
for
$p'\ne p_0$
. Then
$\pi (v)\ne 0$
, but
$\sigma _w(\pi (v))=\rho _w(v)=0$
because
$D(v)\supseteq (p_0^{k_0} {\mathbb Z})^{\mathsf {c}} \in w$
.
Remark 7.4. The equivalence relation
$\mathrel {\equiv ^{\mathrm {s}}_{w}}$
is a congruence with respect to
$\odot $
by [Reference Šobot11, Theorem 5.7(a)]. We leave it to the reader to check that everything above in this section works for
$(\beta {\mathbb Z}, \oplus , \odot )$
, with
$\hat {{\mathbb Z}}$
viewed as a ring, where closed subgroups are replaced by closed ideals, etc.
We already saw several different characterisations of self-divisibility. In order to provide more, we recall a fact from the theory of profinite groups and make an easy observation.
Fact 7.5. If G is a profinite group and
$\sigma \colon \hat {{\mathbb Z}}\to G$
is a surjective homomorphism (of abstract groups), then it is automatically continuous.
Proof sketch
It is enough to show that if U is an open subgroup of G, then
$\sigma ^{-1}(U)$
is open in
$\hat {{\mathbb Z}}$
. By compactness, open subgroups have finite index, hence, it suffices to show that every finite index subgroup of
$\hat {{\mathbb Z}}$
is open. This is in fact true of every topologically finitely generated profinite group by a deep result of Nikolov and Segal [Reference Nikolov and Segal7], but this special case has a quick proof, which we provide for the sake of completeness. Namely, if H has index
$ n$
in
$\hat {{\mathbb Z}}$
, then
$n \hat {{\mathbb Z}}\subseteq H$
, so H can be partitioned into cosets of
$n\hat {{\mathbb Z}}$
, hence it suffices to show that
$n\hat {{\mathbb Z}}$
is open. But
$n \hat {{\mathbb Z}}$
is easily checked to be closed and of finite index, which is equivalent to being open.
Proposition 7.6. If
$u\mathrel {\tilde \mid } v$
and
$v\mathrel {\mid ^{\mathrm {s}}} t$
, then
$u\mathrel {\mid ^{\mathrm {s}}} t$
.
Proof Assume
$u\mathrel {\tilde \mid } v$
and
$v\mathrel {\mid ^{\mathrm {s}}} t$
. Let
$(b, c)\models v\otimes t$
and let
$a\models u$
be such that
$a\mid b$
. Then
$b\mid c$
and thus
$a\mid c$
, but
$\lvert a \rvert \leq \lvert b \rvert $
and thus
$u\mathrel {\mid ^{\mathrm {s}}} t$
by Fact 2.1.
Theorem 7.7. The following are equivalent for
$w\in \beta {\mathbb Z}\setminus \{0\}$
.
-
(1) The ultrafilter
$w$
is self-divisible. -
(2) For all
$B\in w$
there is
$A\in w$
such that for all
$a,a'\in A$
there is
$b\in B$
with
$b\mid \operatorname {gcd}(a,a')$
. -
(3) For all
$B\in w$
there are
$A\in w$
and
$b\in B$
such that
$A\subseteq b \mathbb Z$
. -
(4) For all
$B\in w$
there is
$b\in B$
such that
$b {\mathbb Z}\in w$
. -
(5) For all
$B\in w$
we have
$\{b\in B: b \mathbb Z\in w\}\in w$
. -
(6) For all
$k\in {\mathbb Z}\setminus \{0\}$
we have that
$kw$
is self-divisible. -
(7) There are
$n\ne m$
such that
$w^{\oplus n}\mathrel {\equiv ^{\mathrm {s}}_{w}}w^{\oplus m}$
.Footnote
6
-
(8) For all
$ v$
, if
$w\equiv _{v} 0$
then
$w\mathrel {\equiv ^{\mathrm {s}}_{v}} 0$
. -
(9) If
${}^{\ast }\mathbb {Z}\ni a\models w$
, then
$\{b\in {}^{\ast }\mathbb {Z}: b\mid a\}\subseteq {}^{\ast }D(w)$
. -
(10)
$Z_w$
is closed under
$\oplus $
and, whenever
$v\in \mathrm {MAX}$
, if
$u\oplus v\oplus t\in Z_w$
then
$u\oplus t\in Z_w.$
-
(11)
$Z_w$
is closed under
$\oplus $
and
$Z_w=\pi ^{-1}(\pi (Z_w))$
. -
(12)
$Z_w$
is closed under
$\oplus $
and whether
$w\mathrel {\tilde \mid } u$
only depends on the remainder classes of
$ u$
modulo standard
$ n$
. -
(13) The kernel
$\ker (\sigma _w)$
is closed in
$\hat {{\mathbb Z}}$
. -
(14)
$\beta {\mathbb Z}/\mathord {\mathrel {\equiv ^{\mathrm {s}}_{w}}}$
is a procyclic group with respect to the quotient topology.Footnote
7
-
(15)
$\beta {\mathbb Z}/\mathord {\mathrel {\equiv ^{\mathrm {s}}_{w}}}$
is a profinite group with respect to some topology. -
(16) We have
$(\beta {\mathbb Z}, \oplus )/\mathord {\mathrel {\equiv ^{\mathrm {s}}_{w}}}\cong \prod _{p \in \mathbb P} G_p$
, where
$G_p= {\mathbb Z}_p$
if
$\varphi _w(p)=\omega $
, and
$G_p= {\mathbb Z}/p^{\varphi _w(p)}{\mathbb Z}$
otherwise.
Proof The implication (1)
$\Rightarrow $
(5) is proven by observing that
$w$
is self-divisible if and only if for every
$B\in w$
we have
$D(w)\cap B\neq \emptyset $
, and (5)
$\Rightarrow $
(4)
$\Rightarrow $
(3)
$\Rightarrow $
(2) are immediate.
We now prove (2)
$\Rightarrow $
(1). By assumption and the transfer principle, for every
$B\in w$
there exists
$A\in w$
such that, for every
$a, a'\in {}^{\ast }A$
, there exists
$b\in {}^{\ast }B$
dividing
$\operatorname {gcd}(a, a')$
. Fix two realisations
$a, a'\models w$
. Since for every
$A\in w$
we have
$a, a'\in {}^{\ast }A$
, by assumption for every
$B\in w$
there exists
$b\in {}^{\ast }B$
such that
$b\mid \operatorname {gcd}(a, a')$
. By compactness and saturation we can therefore find
$b\models w$
such that
$b\mid \operatorname {gcd}(a, a')$
and, by Theorem 4.10, this gives (1).
Also (1)
$\Leftrightarrow $
(6) follows from the fact that, for every
$k\in \mathbb {Z}$
, we have
$(a, a')\models w\otimes w$
if and only if
$(ka, ka')\models kw\otimes kw$
. Observe also that, for every
$k\in \mathbb {N}$
, we have
$kw\mathrel {\equiv ^{\mathrm {s}}_{w}} w^{\oplus k}$
. Since (6) implies, by Proposition 7.6, that
$kw\mathrel {\equiv ^{\mathrm {s}}_{w}} 0$
, we conclude that (6)
$\Rightarrow $
(7) by transitivity of
$\mathrel {\equiv ^{\mathrm {s}}_{w}}$
.
We prove (7)
$\Rightarrow $
(1). Assume there are
$(a, b)\models w\otimes w$
such that
$a\nmid b$
. By the transfer principle, there exist
$p\in {}^{\ast }\mathbb {P}$
and
$\alpha \in {}^{\ast }\mathbb {N}$
such that
$p^\alpha \mid a$
but
$p^\alpha \nmid b$
. Notice that
$p$
cannot be finite, since a power of a finite prime dividing
$ a$
must also divide
$ b$
. But then
$p^\alpha \nmid kb$
for every
$k\in \mathbb {Z}$
, and in particular
$kw\mathrel {\not \equiv ^{\mathrm {s}}_{w}} 0$
. Since by (7) there exist
$n<m$
such that
$w^{\oplus n}\mathrel {\equiv ^{\mathrm {s}}_{w}} w^{\oplus m}$
, and as we already observed
$kw\mathrel {\equiv ^{\mathrm {s}}_{w}} w^{\oplus k}$
, we conclude that
$0\mathrel {\equiv ^{\mathrm {s}}_{w}} w^{\oplus (m-n)}\mathrel {\equiv ^{\mathrm {s}}_{w}}(m-n)w$
, a contradiction.
Taking
$v=w$
yields (8)
$\Rightarrow $
(1). Conversely, assuming (1), if
$w\equiv _v 0$
, by Proposition 7.6 and self-divisibility of
$w$
we obtain
$w\mathrel {\equiv ^{\mathrm {s}}_{v}} 0$
, obtaining (8).
To see (1)
$\Leftrightarrow $
(9), notice that
$D(w)=\{n\in \mathbb {Z}: n\mid a\}$
, so
$\{b\in {}^{\ast }\mathbb {Z}: b\mid a\}\subseteq {}^{\ast }D(w)$
if and only if
$a\mid {}^{\ast }a$
, which is the nonstandard characterisation of being self-divisible (see Remark 4.2).
We now prove that (1)
$\Rightarrow $
(12)
$\Rightarrow $
(11)
$\Rightarrow $
(10)
$\Rightarrow $
(1). The equivalence (12)
$\Leftrightarrow $
(11) is immediate from the definitions of
$Z_w$
and
$\pi $
. By Theorem 4.10 and Fact 2.5, if
$w$
is self-divisible then
$Z_w$
is closed under
$\oplus $
, and moreover
$w\mathrel {\tilde \mid } u\Leftrightarrow D(u)\in w$
. Therefore, whether
$w\mathrel {\tilde \mid } u$
only depends on the finite integers dividing
$ u$
, and this yields (1)
$\Rightarrow $
(12).
In order to prove (11)
$\Rightarrow $
(10), recall that, by Lemma 4.8,
$\pi $
is a homomorphism and thus, for every
$u, v, t\in \beta \mathbb {Z}$
, we have
$\pi (u\oplus v\oplus t)=\pi (u)+\pi (v)+\pi (t)$
. If
$v\in \mathrm {MAX}$
, then by Fact 2.7
$\pi (v)$
is the null sequence, so
$\pi (u\oplus v\oplus t)=\pi (u)+\pi (t)=\pi (u\oplus t)$
and the conclusion follows.
In order to prove that (10)
$\Rightarrow $
(1), by Theorem 4.10 it is enough to show that if (10) holds then
$\equiv _w$
is transitive. Let
$u\equiv _w v$
and
$v\equiv _w t$
, i.e.,
$u\ominus v, v\ominus t\in Z_w$
. Since by assumption
$Z_w$
is closed under
$\oplus $
, the ultrafilter
$(u\ominus v)\oplus (v\ominus t)=u\oplus (-v\oplus v)\ominus t$
belongs to
$Z_w$
. But
$-v\oplus v\in \mathrm {MAX}$
, hence by assumption
$u\ominus t\in Z_w$
, or equivalently
$u\equiv _w t$
.
The implication (1)
$\Rightarrow $
(13) was proven in Remark 7.1. To prove (13)
$\Rightarrow $
(1) assume that
$\ker \sigma _w$
is closed. By the characterisation of closed subgroups of
$\hat {{\mathbb Z}}$
(Fact 2.9), there is
$D\subseteq {\mathbb Z}$
of the form
$\bigcap \left (p^{\varphi (p)+1} {\mathbb Z}\right )^{\mathsf {c}}$
such that for all
$u\in \beta {\mathbb Z}$
we have
$\pi (u)\in \ker (\sigma _w)$
if and only if
$D\subseteq D(u)$
. By Proposition 5.17 there is a (possibly principal) self-divisible
$ v$
such that
$D=D(v)$
. Since
$ v$
is self-divisible, by Theorem 4.10, for all
$u\in \beta {\mathbb Z}$
we have
$D(u)\in v\Leftrightarrow v\mathrel {\mid ^{\mathrm {s}}} u\Leftrightarrow v\mathrel {\tilde \mid } u\Leftrightarrow D(v)\subseteq D(u)\Leftrightarrow D\subseteq D(u)\Leftrightarrow w\mathrel {\mid ^{\mathrm {s}}} u\Leftrightarrow D(u)\in w$
, hence
$w$
and
$ v$
contain the same sets of the form
$D(u)$
. Therefore,
$w$
and
$ v$
contain the same
$D(u)^{\mathsf {c}}$
, hence, in particular, the same
$p^k{\mathbb Z}$
, that is,
$D(v)=D(w)$
. But, since
$ v$
is self-divisible,
$D(w)=D(v)\in v$
, hence
$D(w)\in w$
.
Recall that, by Remark 7.1, the topology induced by
$\sigma _w$
coincides with the quotient topology (i.e., the one induced by
$\rho _w$
). Then, that (13)
$\Rightarrow $
(14) follows from the fact that quotients of procyclic groups by closed subgroups are procyclic (see also the characterisation in Footnote 7), and that (14)
$\Rightarrow $
(15) is obvious. Moreover, if
$\beta {\mathbb Z}/\mathord {\mathrel {\equiv ^{\mathrm {s}}_{w}}}$
is profinite with respect to some group topology, by Fact 7.5 the map
$\sigma _w$
is automatically continuous, hence its kernel is closed, proving (15)
$\Rightarrow $
(13).
Finally, (16)
$\Rightarrow $
(15) is clear, and (1)
$\Rightarrow $
(16) is Corollary 7.2.
We take the opportunity to observe that the equivalence of (2) above with point (5) of Theorem 4.10 is a special case of [Reference Luperi Baglini5, Theorem 5.23].
8 Concluding remarks and an open problem
Recall that, by Proposition 3.2,
$\equiv _w$
is always reflexive. In Theorem 3.3 we saw that, whenever
$\equiv _w$
is transitive, then it is automatically symmetric. We were not able to determine whether the converse holds.
Problem 8.1. Are there ultrafilters
$w\in \beta {\mathbb Z} \setminus \{0\}$
such that
$\equiv _w$
is symmetric, but not transitive?
Our investigation of Šobot’s congruence relations
$\equiv _w$
and
$\mathrel {\equiv ^{\mathrm {s}}_{w}}$
led us to introduce self-divisible ultrafilters. The abundance of equivalent forms of self-divisibility (cf. Theorems 4.10 and 7.7) seems to suggest that this and related notions should be investigated further. For instance, one may define
$u/v$
as
$\{A:\{n\in {\mathbb Z}: nA \in v\}\in u\}$
, and observe that
$w$
is self-divisible if and only if
$w/w$
is nonempty, if and only if it is an ultrafilter. We leave it to future work to explore generalisations, for instance, by replacing divisibility with other relations, and applications to areas such as additive combinatorics or Ramsey theory.
Acknowledgements
We thank the anonymous referee for their thorough feedback, that helped to improve the paper.
Funding
M. Di Nasso and R. Mennuni are supported by the Italian research project PRIN 2017: “Mathematical logic: models, sets, computability” Prot. 2017NWTM8RPRIN and are members of the INdAM research group GNSAGA.
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