# Expressive Power of Infinitary Logic and Absolute co-Hopfianity

Mohsen Asgharzadeh  Hakimiyeh, Tehran, Iran<EMAIL_ADDRESS>,
Mohammad Golshani  School of Mathematics,
Institute for Research in Fundamental Sciences (IPM),
P.O. Box: 19395–5746, Tehran, Iran<EMAIL_ADDRESS>and  Saharon Shelah
Einstein Institute of Mathematics, The Hebrew University of Jerusalem,
9190401, Jerusalem, Israel; and
Department of Mathematics, Rutgers University, Piscataway, NJ 08854-8019, USA
https://shelah.logic.at/<EMAIL_ADDRESS>
(Date: 9-05-2022)

###### Abstract.

Recently, Paolini and Shelah have constructed absolutely Hopfian torsion-free
abelian groups of any given size. In contrast, we show that this is not
necessarily the case for absolutely co-Hopfian groups. We apply the infinitary
logic to show a dichotomy behavior of co-Hopfian groups, by showing that the
first beautiful cardinal is the turning point for this property. Namely, we
prove that there are no absolute co-Hopfian abelian groups above the first
beautiful cardinal. An extension of this result to the category of modules
over a commutative ring is given.

###### Key words and phrases:

Absolutely co-Hopfian groups; almost isomorphic; beautiful cardinal;
infinitary logic; elimination of quantifiers; forcing techniques.

###### 2020 Mathematics Subject Classification:

Primary: 20K99, 03C10; Secondary: 03C60, 03C45, 03C75, 20K30.

The third author thanks an individual who wishes to remain anonymous for
generously funding typing services. Research partially supported by the grant
“Independent Theories” NSF-BSF, (BSF 3013005232). References like [Sh:950,
Th0.2=Ly5] mean that the internal label of Th0.2 is y5 in Sh:950. The reader
should note that the version in the author’s website is usually more up-to-
date than the one in arXiv. This is publication number 1246 in Saharon
Shelah’s list.

###### Contents

  1. 1 Introduction
  2. 2 Infinitary languages
    1. 2(A) A review of infinitary languages
    2. 2(B) More on affineness
  3. 3 Additive frames
  4. 4 $\kappa$-algebraic closure
  5. 5 Dichotomy behavior of absolutely co-Hopfian property
  6. 6 A new construction of Hopfian groups

##  1\. Introduction

An abelian group $G$ is called _Hopfian_ (resp. _co-Hopfian_) if its
surjective (resp. injective) endomorphisms are automorphisms. In other words,
co-Hopfian property of groups is, in some sense, dual to the Hopfian groups.
These groups were first considered by Baer in [2], under different names. Hopf
[17] himself showed that the fundamental group of closed two-dimensional
orientable surfaces are hopfian.

There are a lot of interesting research papers in this area. Here, we recall
only a short list of them. Following the book [24], Hopf in 1932, raised the
question as to whether a finitely generated group can be isomorphic to a
proper factor of itself. For this and more observations, see [24]. Beaumont
[4] proved that if $G$ is an abelian group of finite rank all of whose
elements have finite order, then $G$ has no proper isomorphic subgroups.
Kaplansky [18] extended this to modules over a commutative principal ideal
ring $R$ such that every proper residue class ring of $R$ is finite. Beaumont
and Pierce [3] proved that if $G$ is co-Hopfian, then the torsion part of $G$
is of size at most continuum, and further that $G$ cannot be a $p$-groups of
size $\aleph_{0}$. This naturally left open the problem of the existence of
co-Hopfian $p$-groups of uncountable size $\leq 2^{\aleph_{0}}$, which was
later solved by Crawley [9] who proved that there exist $p$-groups of size
$2^{\aleph_{0}}$. One may attempt to construct (co-)Hopfian groups of large
size by taking a huge direct sum of (co-)Hofian groups. In this regard,
Baumslag [7] asked when is the direct sum of two (co-)Hopfian groups again
(co)-Hopfian? Corner [8] constructed two torsion-free abelian Hopfian groups
which have non-Hopfian direct sum. See [15], for more on this and its
connections with the study of algebraic entropy.

Despite its long history, only very recently the problem of the existence of
uncountable (co-)Hopfian abelian groups was solved, see [1] and [22].

The usual construction of such groups is not absolute, in the sense that they
may loose their property in some generic extension of the universe, and the
problem of giving an explicit and constructive construction of such groups has
raised some attention in the literature. For example, while it is known from
the work of Shelah [25] that there are indecomposable abelian groups of any
infinite size, the problem of the existence of arbitrary large absolutely
indecomposable groups is still open, see [21]. It is worth noticing that
indecomposability implies the Hopfian property.

In order to be more explicit, call a group $G$ _absolutely co-Hopfian_ if it
is co-Hopfian in any further generic extension of the universe. Similarly, one
may define _absolutely Hopfian_ groups. An open problem in this area was as
follows:

######  Problem 1.1.

  * (i)

Is it possible to construct absolutely Hopfian torsion-free groups of a given
size?

  * (ii)

Is it possible to construct absolutely co-Hopfian torsion-free groups of a
given size?

Recently, Paolini and Shelah [22, Theorem 1.3] constructed absolutely Hopfian
torsion-free groups of any given size $\lambda$, thereby confirming Problem
1.1(i) in positive direction. On the one hand, and as Hopfian and co-Hopfian
groups are dual to each ether, one may predict that there is a connection
between Problem 1.1 (i) and (ii). But, any such dual functor, may enlarge or
collapse the cardinality of the corresponding groups, hence we can not use the
ideas behind the duality to answer Problem 1.1(ii). For example, Braun and
Strüngmann [6] showed that the existence of infinite abelian $p$-groups of
size $\aleph_{0}<|G|<2^{\aleph_{0}}$ of the following types are independent of
ZFC:

  * (a)

both Hopfian and co-Hopfian,

  * (b)

Hopfian but not co-Hopfian,

  * (c)

co-Hopfian but not Hopfian.

Also, they proved that the above three types of groups of size
$2^{\aleph_{0}}$ exist in ZFC.

On the other hand, there are some partial positive results in the direction of
Problem 1.1 (ii). Here, we recall some of these. In [29], Shelah studied and
coined the concept of a _beautiful cardinal_ , denoted by
$\kappa_{\rm{beau}}$, which is a kind of Ramsey cardinal (see Definition
3.12). This cardinal has an essential role in the study of absolutely co-
Hopfian groups. Indeed, according to [16], for any infinite cardinal
$\lambda<\kappa_{\rm{beau}}$, there is an absolutely endo-rigid abelian group,
but if $|G|\geq\kappa_{\rm{beau}}$, then by [12] it has non-trivial
monomorphisms. Let us state the remaining part of Problem 1.1:

######  Problem 1.2.

Is it possible to construct absolutely co-Hopfian torsion-free groups of size
$\geq\kappa_{\rm{beau}}$?

The Hopfian and co-Hopfian property easily can be extended to the context of
modules over commutative rings and even to the context of sheaves over
schemes. However, compared to the case of abelian groups, and up to our
knowledge, there are very few results for modules. For example, a funny result
of Vasconcelos [31, 1.2] says that any surjective $f:M\to M$ is an
isomorphism, where $M$ is a noetherian module over a commutative and
noetherian ring $R$. As a geometric sample, let $X$ be an algebraic variety
over an algebraically closed field. If a morphism $f:X\to X$ is injective then
a result of Ax and Grothendieck indicates that $f$ is bijective, see Jean-
Pierre Serre’s exposition [23].

In contrast to the the case of Hopfian groups, we prove the following
dichotomy property of co-Hopfian groups. Indeed, we prove a more general
result in the context of additive $\tau$-models (see Definition 2.3), which
includes in particular the cases of abelian groups and $R$-modules, for an
arbitrary commutative ring $R$:

######  Theorem 1.3.

The following assertions are valid:

  1. (A)

If $M$ is an abelian group of cardinality greater or equal
$\kappa:=\kappa_{\rm{beau}}$, then $M$ is not absolutely co-Hopfian, indeed,
after collapsing the size of $M$ into $\omega$, there is a one-to-one
endomorphism $\varphi\in\rm{End}(M)$ which is not onto.

  2. (B)

If $M$ is an $R$-module of cardinality greater or equal
$\kappa=\kappa_{\rm{beau}}(R,\aleph_{0})$, then $M$ is not absolutely co-
Hopfian.

  3. (C)

If $M$ is an additive $\tau$-model of cardinality greater or equal
$\kappa=\kappa_{\rm{beau}}(\tau)$, then $M$ is not absolutely co-Hopfian.

Putting things together, we conclude that for every infinite cardinal
$\lambda,$ there is an absolutely co-Hopfian abelian group of size $\lambda$
if and only if $\lambda$ is less than the first beautiful cardinal.

The organization of this paper is as follows.

Szmielew [30] developed first order theory of abelian groups, see also [28]
for further discussions. In Section 2 we review infinitary languages and
develop some parts of abelian group theory in this context. For this, we use
the concept of $\theta$-models from [27]. We also introduce some sub-languages
of the infinitary languages, which play some role in our later investigation.

For Section 3, let us fix a pair $(\lambda,\theta)$ of regular cardinals and
let $\kappa$ be as Theorem 1.3. The new object studied here is called the
general frame and its enrichment the _additive frame_. Such a frame is of the
form

$\mathbf{f}:=(M,\mathscr{L},\lambda,\kappa,\theta,\Omega),$

where $M$ comes from Theorem 1.3, and it has an additive $\tau_{M}$-model of
cardinality $\geq\kappa$ and $\mathscr{L}$ is a class of certain formulas in
the vocabulary $\tau_{M}$. For more details, see Definition 3.1. The main
result of Section 3 is Theorem 3.14. This gives us an additive frame
$\mathbf{f}$.

Section 4 is about the concept of _algebraic closure_ in a frame (see
Definition 4.1). This enables us to improve Theorem 3.14, which is needed in
the squeal.

In Section 5 we put all things together and present the proof of Theorem 1.3.
Let $\chi:=|M|\geq\kappa$, and let $\mathbb{P}:=\rm{Col}(\aleph_{0},\chi)$.
Forcing with $\mathbb{P}$ enables us to collapse $|M|$ into $\aleph_{0}$,
i.e., for any $\mathbb{P}$-generic filter $G_{\mathbb{P}}$ over $V$, we have

$V[G_{\mathbb{P}}]\models$“$M$ is countable”.

We show in $V[G_{\mathbb{P}}]$, there exists a 1-1 map $\pi:M\to M$ which is
not surjective.

Finally, as another application of infinitary logic, we present a new
construction of Hopfian groups.

We close the introduction by noting that all groups (resp. rings) are abelian
(resp. commutative), otherwise specialized, and our notation is standard and
follows that in Fuchs books [14] and [13] and Eklof-Mekler [10].

##  2\. Infinitary languages

In the first subsection, we briefly review infinitary languages and the
concept of additive $\theta$-models, introduced in [27]. In the second
subsection, we present basic properties of affine subsets, i.e., ones closed
under $x-y+z$. The main result is Theorem 2.22.

###  2(A). A review of infinitary languages

In this subsection we briefly review the infinitary logic, and refer to [5]
and [27] for more information.

######  Convention 2.1.

Given a model $M$, by $\tau_{M}$ we mean the language (or vocabulary) of the
model $M$.

######  Notation 2.2.

  1. (A)

By $\mathrm{AB}$ we mean the class of abelian groups.

  2. (B)

Given a vocabulary $\tau$ which contains two place functions $+,-$, we define
the affine operation $\mathrm{Affine}({x},{y},{z})$ as the three place
function $\mathrm{Affine}({x},{y},{z}):={x}-{y}+{z}$.

######  Definition 2.3.

Let $M$ be a model of vocabulary $\tau_{M}$.

  1. (A)

We say $M$ is an _additive $\theta$-model_ when:

    1. (a)

the two place function symbols $+,-$ and the constant symbol $0$ belong to
$\tau_{M},$

    2. (b)

$G_{M}=(|M|,+^{M},-^{M},0^{M})\in\mathrm{AB}$,

    3. (c)

$R^{M}$ is a subgroup of ${}^{n}(G_{M})$, for any predicate symbol
$R\in\tau_{M}$ with $\rm{arity(R)}=n$,

    4. (d)

$F^{M}$ is a homomorphism from ${}^{n}(G_{M})$ into $G_{M}$, for any function
symbol $F\in\tau_{M}$ with arity $n,$

    5. (e)

$\tau_{M}$ has cardinality $\leq\theta.$

  2. (B)

For an additive $\theta$-model $M$, we say $X\subseteq{M}$ is affine if $X$ is
closed under the affine operation $\mathrm{Affine}({x},{y},{z})$. In other
words, ${a}-{b}+{c}\in X$ provided that ${a},{b},{c}\in X$.

  3. (C)

We say $M$ is an affine $\theta$-model provided:

    1. (a)

we do not necessarily have $+,-,0$ in the vocabulary, but only the three place
function $\mathrm{Affine}(x,y,z)$,

    2. (b)

if $R\in\tau_{M}$ is an $n$-place predicate and $\overline{a}_{l}=\langle
a_{l,i}:i<n\rangle\in R^{M}$ for $l=0,1,2$ and

$\overline{b}:=\mathrm{Affine}(\overline{a}_{0},\overline{a}_{1},\overline{a}_{2})=\big{\langle}\mathrm{Affine}(a_{0,i},a_{1,i},a_{2,i}):i<n\big{\rangle},$

then $\overline{b}\in R^{M}$,

    3. (c)

for any $n$-place function symbol $F\in\tau_{M}$ and $\overline{a}_{l}=\langle
a_{l,i}:i<n\rangle\in{}^{n}M$, for $l=0,1,2$, we have

$F^{M}(\mathrm{Affine}(\overline{a}_{0},\overline{a}_{1},\overline{a}_{2}))=\mathrm{Affine}(F^{M}(\overline{a}_{0}),F^{M}(\overline{a}_{1}),F^{M}(\overline{a}_{2})),$

    4. (d)

$\tau_{M}$ has cardinality $\leq\theta.$

  4. (D)

Suppose $M$ is an affine $\theta$-model. We say $M$ is truly affine provided
for some fixed $a\in M$ and for the following interpretation

     * •

$x+y:=\mathrm{Affine}(x,a,y)=x-a+y,$

     * •

$x-y:=\mathrm{Affine}(x,y,a)=x-y+a,$

     * •

$0:=a$,

then we get an abelian group, and hence an additive $\theta$-model.

We may omit $\theta$ if it is clear from the context.

######  Remark 2.4.

i) A natural question arises: Is any affine $\theta$-model truly affine? Not
necessarily this holds, see Example 2.5, below.

ii) More generally, we can replace $\\{+,-,\mathrm{Affine}\\}$ for a set
$\tau_{f}$ of beautiful function from [26]. The corresponding result holds in
this frame.

######  Example 2.5.

Let $G$ be an abelian group, $H$ be a proper subgroup of it and $a\in
G\setminus H.$ Define $M$ as follows:

  * •

the universe of $M$ is $a+H,$

  * •

$\tau_{M}:=\\{+,-,\mathrm{Affine}\\}$,

  * •

$+^{M}$ and $-^{M}$ are $+^{G}\restriction M$ and $-^{G}\restriction M$
respectively,

  * •

$\mathrm{Affine}^{M}:=\mathrm{Affine}^{G}\restriction M$, where
$\mathrm{Affine}^{G}=\\{x-y+z:x,y,z\in G\\}$.

Then the following two assertions hold:

  * a)

$M$ is an affine $\aleph_{0}$-model, isomorphic to $H$.

  * b)

$M$ is not an abelian group.

######  Definition 2.6.

(1) We say a class $K$ of models is an _additive $\theta$-class_, when $M$ is
an additive $\theta$-model for all $M\in K$, and

$\tau_{M}=\tau_{N}\quad\forall M,N\in K.$

We denote the resulting common language by $\tau_{K}.$

(2) Similarly, one can define affine $\theta$-classes.

######  Hypothesis 2.7.

Let $\Omega$ be a set of cardinals with $1\in\Omega$ and members of
$\Omega\setminus\\{1\\}$ are infinite cardinals.

######  Notation 2.8.

  1. (A)

By $\overline{x}_{[u]}$ or $\overline{x}_{u}$ we mean $\langle
x_{\alpha}:\alpha\in u\rangle.$ So, with no repetition.

  2. (B)

Suppose $\varphi(\overline{x}_{u})$ is a formula. By $\varphi(M)$ we mean
$\\{\overline{a}\in{}^{u}M:M\models\varphi[\overline{a}]\\}.$

  3. (C)

For a formula $\varphi(\overline{x}_{[u]},\overline{y}_{[v]})$ and
$\overline{b}\in{}^{v}M,$ we let

$\varphi(M,\overline{b}):=\big{\\{}\overline{a}\in{}^{u}M:M\models\varphi[\overline{a},\overline{b}]\big{\\}}.$

  4. (D)

Given a sequence $t$, by $\rm{lg}(t)$ we mean the length of $t$.

######  Definition 2.9.

Suppose $\kappa$ and $\mu$ are infinite cardinals, which we allow to be
$\infty$. The infinitary language $\mathcal{L}_{\mu,\kappa}(\tau)$ is defined
so as its vocabulary is the same as $\tau,$ it has the same terms and atomic
formulas as in $\tau,$ but we also allow conjunction and disjunction of length
less than $\mu$, i.e., if $\phi_{j},$ for $j<\beta<\mu$ are formulas, then so
are $\bigvee_{j<\beta}\phi_{j}$ and $\bigwedge_{j<\beta}\phi_{j}$. Also,
quantification over less than $\kappa$ many variables (i.e., if
$\phi=\phi((v_{i})_{i<\alpha})$, where $\alpha<\kappa$, is a formula, then so
are $\forall_{i<\alpha}v_{i}\phi$ and $\exists_{i<\alpha}v_{i}\phi$).

Note that $\mathcal{L}_{\omega,\omega}(\tau)$ is just the first order logic
with vocabulary $\tau.$ Given $\kappa$, $\mu$ and $\tau$ as above, we are
sometimes interested in some special formulas from
$\mathcal{L}_{\mu,\kappa}(\tau)$.

######  Definition 2.10.

Suppose $\kappa$ and $\lambda$ are infinite cardinals or possibly $\infty$. We
define the logic $\mathscr{L}_{\lambda,\kappa,\Omega}$ as follows:

  1. (A)

For a vocabulary $\tau$, the language
$\mathscr{L}_{\lambda,\kappa,\Omega}(\tau)$ is defined as the set of formulas
with $<\theta$ free variables (without loss of generality they are subsets of
$\\{x_{\zeta}:\zeta<\theta\\}$, see Discussion 2.11) which is the closure of
the set of basic formulas , i.e., atomic and the negation of atomic formulas,
under:

     * (a)

conjunction of $<\lambda$ formulas,

     * (b)

disjunction of $<\lambda$ formulas,

     * (c)

For any $\sigma\in\Omega$,

       * ($c_{1}$)

$\varphi(\overline{x}):=(\exists^{\sigma}\overline{x}^{\prime})\psi(\overline{x},\overline{x}^{\prime})$,
or

       * ($c_{2}$)

$\varphi(\overline{x}):=(\forall^{\sigma}\overline{x}^{\prime})\psi(\overline{x},\overline{x}^{\prime})$,

where $\psi(\overline{x},\overline{x}^{\prime})$ is a formula. We usually omit
$\sigma$, if $\sigma=1$.

We usually omit $\Omega$ if it is clear from the context.

  2. (B)

Satisfaction is defined as usual, where for the formulas
$\varphi(\overline{x}):=(\exists^{\sigma}\overline{x}^{\prime})\psi(\overline{x},\overline{x}^{\prime})$
and
$\varphi(\overline{x}):=(\forall^{\sigma}\overline{x}^{\prime})\psi(\overline{x},\overline{x}^{\prime})$,
it is defined as:

     * (a)

If
$\varphi(\overline{x}):=(\exists^{\sigma}\overline{x}^{\prime})\psi(\overline{x},\overline{x}^{\prime})$,
$M$ is a $\tau$-model, and $\overline{a}\in{}^{\lg(\overline{x})}M$, then

$M\models\varphi(\overline{a})$

if and only if there are
$\overline{b}_{\varepsilon}\in{}^{\lg(\overline{x}^{\prime})}M$ for all
$\varepsilon<\sigma$ pairwise distinct such that
$M\models\psi(\overline{a},\overline{b}_{\varepsilon})$ for all
$\varepsilon<\sigma$.

     * (b)

If
$\varphi(\overline{x}):=(\forall^{\sigma}\overline{x}^{\prime})\psi(\overline{x},\overline{x}^{\prime})$,
then

$M\models\varphi(\overline{x})\iff
M\models\neg\big{[}\exists^{\sigma}\overline{x}^{\prime}\neg\big{(}\psi(\overline{x},\overline{x}^{\prime})\big{)}\big{]}.$

Note that $\neg(\psi(\overline{x},\overline{x}^{\prime}))$ is not necessarily
in $\mathscr{L}_{\lambda,\kappa,\Omega}(\tau)$.

######  Discussion 2.11.

Given a formula $\varphi$ in $\mathcal{L}_{\infty,\theta}(\tau)$ with free
variables $\overline{x}_{\varphi},$ we can always assume that
$\overline{x}_{\varphi}=\langle x_{\zeta}:\zeta\in u_{\varphi}\rangle,$ for
some $u_{\varphi}\in[\theta]^{<\theta}$. The key point is that if
$\varphi=\varphi(\overline{x}),$ where $\overline{x}=\langle
x_{\zeta}:\zeta\in w\rangle,$ where $w$ is a set of ordinals of size less than
$\theta$, and if $f:w\leftrightarrow u$ is a bijection where
$u\in[\theta]^{<\theta}$, and

$\psi(\overline{x})\equiv\mathrm{Sub}_{f}^{\overline{x}}(\varphi),$

where $\mathrm{Sub}_{f}^{\overline{x}}(\varphi)$ is essentially the formula
obtained from $\varphi$ by replacing the variable $x_{\zeta}$ by
$x_{f(\zeta)}$, then if $\bar{a}\in$${}^{w}M$, $\bar{b}\in$${}^{u}M$ and
$a_{\zeta}=b_{f(\zeta)}$, for $\zeta\in w$, then

$M\models\varphi(\bar{a})\iff M\models\psi(\bar{b}).$

We can similarly assume that all bounded variables are from
$\\{x_{i}:i<\theta\\}$.

######  Convention 2.12.

In what follows, saying closed under $\exists$ (resp. $\forall$) means under
all $\exists^{\sigma}$ (resp. $\forall^{\sigma}$).

In the next definition, we consider some classes of infinitary formulas that
we will work with them latter.

######  Definition 2.13.

Suppose $\theta$ is an infinite cardinal, or $\infty,$ and suppose $\tau$ is a
language. Here, we collect some infinitary sub-classes of the language
$\mathscr{L}_{\infty,\theta}(\tau)$:

  1. (A)

$\mathscr{L}_{\infty,\theta}^{\rm{cop}}(\tau)$ is the class of conjunction-
positive formulas, i.e., the closure of atomic formulas under
$\bigwedge,\exists,\forall$.

  2. (B)

$\mathscr{L}_{\infty,\theta}^{\rm{cpe}}(\tau)$ is the class of conjunction-
positive existential formulas, i.e., the closure of atomic formulas under
$\bigwedge$ and $\exists$.

  3. (C)

$\mathscr{L}_{\infty,\theta}^{\rm{co}}(\tau)$ is the closure of atomic
formulas and $x_{i}\neq x_{j}$ under $\bigwedge$, $\exists$ and $\forall$.

  4. (D)

$\mathscr{L}_{\infty,\theta}^{\rm{ce}}(\tau)$ is the closure of atomic
formulas and $x_{i}\neq x_{j}$ under $\bigwedge$ and $\exists$.

We shall use freely the following simple fact.

######  Fact 2.14.

$\mathscr{L}_{\infty,\theta}^{\rm{co}}(\tau)\supseteq\mathscr{L}_{\infty,\theta}^{\rm{cop}}(\tau)\cup\mathscr{L}_{\infty,\theta}^{\rm{ce}}(\tau)\supseteq\mathscr{L}_{\infty,\theta}^{\rm{cop}}(\tau)\cap\mathscr{L}_{\infty,\theta}^{\rm{ce}}(\tau)\supseteq\mathscr{L}_{\infty,\theta}^{\rm{cpe}}(\tau)$.

The following lemma is easy to prove.

######  Lemma 2.15.

Assume $M$ is an additive $\theta$-model, $\tau=\tau_{M}$ and
$\varphi(\overline{x}_{u})\in\mathscr{L}_{\infty,\infty}(\tau)$ with
$\varepsilon=\rm{lg}(\overline{x})$. The following assertions are valid:

  1. (A)

If $\varphi(\overline{x}_{u})\in\mathscr{L}_{\infty,\infty}^{\rm{cop}}(\tau)$,
then $\varphi(M)$ is a sub-group of ${}^{u}M.$

  2. (B)

If $\varphi(\overline{x}_{u})\in\mathscr{L}^{\rm{cpe}}_{\infty,\infty}(\tau)$,
$f\in\rm{End}(M)$ and $M\models\varphi[\overline{a}]$, then
$M\models\varphi[f(\overline{a})].$

  3. (C)

If $\varphi(\overline{x}_{u})\in\mathscr{L}^{\rm{cpe}}_{\infty,\theta}(\tau)$,
$M,N$ are $\tau$-models and $f:M\to N$ is a homomorphism, then $f$ maps
$\varphi(M)$ into $\varphi(N).$

  4. (D)

If $\varphi(\overline{x}_{u})\in\mathscr{L}^{\rm{ce}}_{\infty,\theta}(\tau)$,
$M,N$ are $\tau$-models and $f:M\to N$ is a 1-1 homomorphism, then $f$ maps
$\varphi(M)$ into $\varphi(N).$

  5. (E)

If $\varphi(\overline{x}_{u})\in\mathscr{L}^{\rm{co}}_{\infty,\theta}(\tau)$,
$M,N$ are $\tau$-models and $f:M\to N$ is a bijection, then $f$ is an
isomorphism from $\varphi(M)$ onto $\varphi(N).$

  6. (F)

Assume $\psi(\bar{y})$ is obtained from $\varphi(\bar{x})$ by adding dummy
variables, permuting the variables and substitution not identifying variables.
Then

$\psi(\bar{y})\in\mathscr{L}^{\ast}_{\infty,\theta}(\tau)\iff\varphi(\bar{x})\in\mathscr{L}^{\ast}_{\infty,\theta}(\tau),$

where $\ast\in\\{\rm{cop},\rm{cpe},\rm{ce},\rm{co}\\}$.

###### Proof.

The proof is by induction on the complexity of the formulas. For completeness,
we sketch the proof. If the formula is an atomic formula, then it is evident
that all of the above items are satisfied. It is also easy to see that each
item is preserved under $\bigwedge$, in the sense that if
$\psi=\bigwedge_{i\in I}\varphi_{i}$ is well-defined and the lemma holds for
each $\varphi_{i}$, then it holds for $\psi$.

We now consider the case where
$\psi(\bar{x})=(\exists^{\sigma}\bar{y})\varphi(\bar{x},\bar{y})$, and assume
the induction hypothesis holds for $\varphi.$ We consider each clause
separately, assuming in each case, the formula $\varphi$ is in the assumed
language.

  1. Clause (1): Suppose $\varphi(M)$ is a subgroup of ${}^{\rm{lg}(\bar{x})+\rm{lg}(\bar{y})}M$. We show that $\psi(M)$ is a subgroup of ${}^{\rm{lg}(\bar{x})}M$. To see this, let $\bar{a}_{0},\bar{a}_{1}\in\psi(M)$. Then for some $\bar{b}_{0}$ and $\bar{b}_{1}$ we have

$M\models``\varphi[\bar{a}_{0},\bar{b}_{0}]\emph{ and
}\varphi[\bar{a}_{1},\bar{b}_{1}]^{\prime\prime}.$

By induction, $M\models$$\varphi[\bar{a}_{0}$ \- $\bar{a}_{1},\bar{b}_{0}$ \-
$\bar{b}_{1}]$, hence

$M\models\psi[\bar{a}_{0}-\bar{a}_{1}].$

Thus $\bar{a}_{0}$-$\bar{a}_{1}\in\psi(M)$.

  2. Clause (2): Suppose $M\models$$\psi[\bar{a}]$. Then for some $\bar{b}$, we have $M\models\varphi[\bar{a},\bar{b}]$. By the induction, $M\models\varphi[f(\bar{a}),f(\bar{b})]$, and hence $M\models\psi[f(\bar{a})]$, as requested.

  3. Clause (3): As in clause (2), we can show that if $M\models$$\psi[\bar{a}]$, then $N\models$$\psi[f(\bar{a})]$, and this gives the required result.

  4. Clause (4): As in clause (3). The assumption of $f$ being 1-1 is used to show that if $x_{i}\neq x_{j}$, then $f(x_{i})\neq f(x_{j})$.

  5. Clause (5): As in clause (4).

  6. Clause (6): This is easy.

Finally, suppose that
$\psi(\bar{x})=(\forall^{\sigma}\bar{y})\varphi(\bar{x},\bar{y})$, and assume
the induction hypothesis holds for $\varphi$. We only have to consider items
(1), (5) and (6).

  1. Clause (1): Suppose $\varphi(M)$ is a subgroup of ${}^{\rm{lg}(\bar{x})+\rm{lg}(\bar{y})}M$. We show that $\psi(M)$ is a subgroup of ${}^{\rm{lg}(\bar{x})}M$. To see this, let $\bar{a}_{0},\bar{a}_{1}\in\psi(M)$. We have to show that $\bar{a}_{0}$-$\bar{a}_{1}\in\psi(M)$. Thus let $\bar{b}\in$${}^{\rm{lg}(\bar{y})}M$. By the induction hypothesis,

$M\models``\varphi[\bar{a}_{0},\bar{b}]\emph{ and
}\varphi[\bar{a}_{1},\bar{0}]^{\prime\prime}.$

Thanks to induction, $M\models$$\varphi[\bar{a}_{0}$ \- $\bar{a}_{1},\bar{b}$
\- $\bar{0}]$. As this holds for all $\bar{b}$, we have

$M\models\psi[\bar{a}_{0}-\bar{a}_{1}].$

Thus $\bar{a}_{0}$-$\bar{a}_{1}\in\psi(M)$, as requested.

  2. Clause (5): As before, we can easily show that $f$ maps $\psi(M)$ into $\psi(N).$ To see it is onto, let $\bar{c}\in\psi(N)$. Then $N\models\psi(\bar{c})$. As $f$ is onto, for some $\bar{a}$ we have $\bar{c}=f(\bar{a})$. We have to show that $\bar{a}\in\psi(M)$. Thus let $\bar{b}\in$${}^{\rm{lg}(\bar{y})}M$. Then $\bar{d}=f(\bar{b})\in$${}^{\rm{lg}(\bar{y})}N$, and by our assumption, $N\models\varphi(\bar{c},\bar{d})$. As $f$ is an isomorphism, $M\models\varphi(\bar{a},\bar{b})$. As $\bar{b}$ was arbitrary, $M\models\psi(\bar{a})$, i.e., $\bar{a}\in\psi(M)$.

  3. Clause (6): This is easy.

The lemma follows. ∎

Let us repeat the above result in the context of $R$-modules:

######  Corollary 2.16.

Let $M$ be an $R$-module.

  1. (A)

If $\varphi(\overline{x}_{u})\in\mathscr{L}_{\infty,\theta}^{\rm{{cpe}}},$
then $\varphi(M)$ is an abelian subgroup of $({}^{u}M,+).$

  2. (B)

Similar result holds for formulas
$\varphi(\overline{x}_{u})\in\mathscr{L}_{\infty,\theta}^{\rm{{cop}}}.$

Furthermore, if $R$ is commutative, then in the above, $\varphi(M)$ becomes a
sub-module of $({}^{u}M,+).$

######  Remark 2.17.

If $R$ is not commutative, then $\varphi(M)$ is not necessarily a submodule.
To see this, suppose $a,b,c\in R$ are such that $abc\neq bac,$ and suppose $M$
is a left $R$-module. Define $F_{a}:M\to M$ as $F_{a}(x)=ax$. Now note that
$(c,ac)\in F_{a}(M).$ If $F_{a}(M)$ is a sub-module, then we must have
$(bc,bac)\in F_{a}(M).$ By definition, $(bc,bac)=F_{a}(x)=(x,ax)$ for some
$x\in M$. It then follows that $x=bc$ and hence

$bac=ax=abc,$

which contradicts $abc\neq bac.$

###  2(B). More on affineness

In this subsection we try replacing subgroups by affine subsets. The main
result of this subsection is Proposition 2.22. First, we fix the hypothesis
and present the corresponding definitions. Here, affinity demand relates only
to the formulas, not the content.

######  Hypothesis 2.18.

  1. (A)

$R$ is a ring,

  2. (B)

$M$ is an $R$-module,

  3. (C)

$\kappa,\theta$ are as in Definition 3.1.

######  Definition 2.19.

Let $\mathrm{Affine}_{1}$ be the set of all formulas
$\varphi(\overline{x})\in\mathscr{L}_{\infty,\theta}(\tau_{M})$ so that
$\rm{lg}(\overline{x})<\theta$ and $\varphi(M)$ is closed under
$\overline{x}-\overline{y}+\overline{z}$. In other words,
$\overline{a}-\overline{b}+\overline{c}\in\varphi(M)$ provided that
$\overline{a},\overline{b},\overline{c}\in\varphi(M)$.

We now define another class $\mathrm{Affine}_{2}$ of formulas of
$\mathscr{L}_{\infty,\theta}(\tau_{M})$, and show that it is included in
$\mathrm{Affine}_{1}$. To this end, we first make the following definition.

######  Definition 2.20.

Suppose $\alpha_{*}$ is an ordinal. Let $\varphi(\overline{x},\overline{y})$
and
$\overline{\psi}(\overline{x},\overline{y})=\langle\psi_{\alpha}(\overline{x},\overline{y}):\alpha<\alpha_{*}\rangle$
be a sequence of formulas from $\mathscr{L}_{\infty,\theta}(\tau_{M})$. Let
$\overline{b}\in{}^{\operatorname{lg}(\overline{x})}{M}$ and
$\overline{a}\in{}^{\operatorname{lg}(\overline{y})}{M}$. Then we set

  1. (A)

$\mathrm{set}_{\overline{\psi}}(\overline{b},\overline{a})$ stands for the
following set

$\mathrm{set}_{\overline{\psi}}(\overline{b},\overline{a}):=\big{\\{}\alpha\in\alpha_{*}:(\overline{b}^{\frown}\overline{a}\in\psi_{\alpha}(M))\big{\\}}.$

  2. (B)

By $\operatorname{Set}_{\varphi,\overline{\psi}}(\overline{a})$ we mean

$\operatorname{Set}_{\varphi,\overline{\psi}}(\overline{a}):=\big{\\{}u\subseteq\alpha_{*}:\text{for
some}\ \overline{c}\in\varphi(M,\overline{a})\ \text{we have}\
u=\mathrm{set}_{\overline{\psi}}(\overline{c},\overline{a})\big{\\}}.$

  3. (C)

By $\operatorname{inter}_{\varphi,\overline{\psi}}(\overline{a})$ we mean

$\bigg{\\{}(w_{0},w_{1}):w_{0}\subseteq w_{1}\subseteq\alpha_{*}\ \text{and
}\exists\
u_{0},u_{1}\in\operatorname{Set}_{\varphi,\overline{\psi}}(\overline{a})\
\text{s.t.}\ w_{1}\subseteq u_{1}\ \text{and}\ u_{0}\cap
w_{1}=w_{0}\bigg{\\}}.$

In particular, we have the following flowchart:

$\textstyle{\alpha_{*}}$$\textstyle{u_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\subseteq}$$\textstyle{u_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\subseteq}$$\textstyle{w_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\subseteq}$$\scriptstyle{\subseteq}$$\textstyle{w_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\subseteq}$

.

We are now ready to define the class $\mathrm{Affine}_{2}$ of formulas:

######  Definition 2.21.

Let $\mathrm{Affine}_{2}$ be the closure of the set of atomic formulas by:

  1. (A)

arbitrary conjunctions,

  2. (B)

existential quantifier $\exists\overline{x},$ and

  3. (C)

suppose for a given ordinal $\alpha_{*}$, the formulas
$\varphi(\overline{x},\overline{y}),\langle\psi_{\alpha}(\overline{x},\overline{y}):\alpha<\alpha_{*}\rangle$
are taken from $\mathrm{Affine}_{2}$ such that
$\varphi(\overline{x},\overline{y})\geq\psi_{\alpha}(\overline{x},\overline{y})$
for all $\alpha<\alpha_{*}$. Also suppose that

$\Upsilon\subseteq\\{(w_{0},w_{1}):w_{0}\subseteq
w_{1}\subseteq\alpha_{*}\\}.$

Then
$\vartheta(\overline{y})=\Theta_{\varphi,\overline{\psi},\Upsilon}(\overline{y})\in\mathrm{Affine}_{2},$
where $\vartheta(\overline{y})$ is defined such that

$M\models\vartheta[\overline{a}]\iff\Upsilon\subseteq\operatorname{inter}_{\varphi,\overline{\psi}}(\overline{a}).$

The main result of this section is the following.

######  Proposition 2.22.

Adopt the previous notation. Then
$\mathrm{Affine}_{2}\subseteq\mathrm{Affine}_{1}.$

###### Proof.

We prove the theorem in a sequence of claims. We proceed by induction on the
complexity of the formula $\vartheta$ that if
$\vartheta\in\mathrm{Affine}_{2},$ then $\vartheta\in\mathrm{Affine}_{1}$.
This is clear if $\vartheta$ is an atomic formula. Suppose
$\vartheta=\bigwedge_{i\in I}\vartheta_{i}$, and the claim holds for all
$\vartheta_{i},i\in I$. It is then clear from inductive step that
$\vartheta\in\mathrm{Affine}_{1}$ as well. Similarly, if
$\vartheta=\exists\overline{x}\varphi(\overline{x}),$ and if the claim holds
for $\varphi(\overline{x})$, then clearly it holds for $\vartheta$.

Now suppose that $\alpha_{*}$ is an ordinal,
$\varphi(\overline{x},\overline{y}),\langle\psi_{\alpha}(\overline{x},\overline{y}):\alpha<\alpha_{*}\rangle$
are in $\mathrm{Affine}_{2}$, such that
$\varphi(\overline{x},\overline{y})\geq\psi_{\alpha}(\overline{x},\overline{y})$
for all $\alpha<\alpha_{*}.$ Also, suppose that

$\Upsilon\subseteq\\{(w_{0},w_{1}):w_{0}\subseteq
w_{1}\subseteq\alpha_{*}\\}.$

Assume by the induction hypothesis that the formulas
$\varphi(\overline{x},\overline{y})$ and
$\psi_{\alpha}(\overline{x},\overline{y})$, for $\alpha<\alpha_{*}$, are in
$\mathrm{Affine}_{1}.$ We have to show that
$\vartheta(\overline{y})=\Theta_{\varphi,\overline{\psi},\Upsilon}(\overline{y})\in\mathrm{Affine}_{1}$
as well.

Now, we bring the following claim:

######  Claim 2.23.

Adopt the above notation. Assume
$\overline{a}_{l}\in{}^{\rm{lg(\overline{y})}}M$ for $l=0,1,2,3$ and
$\overline{a}_{3}=\overline{a}_{0}-\overline{a}_{1}+\overline{a}_{2}.$ If
$u_{j}\in\rm{Set}(\overline{a}_{j})$ for $j=0,1$ and $u=u_{0}\cap u_{1}$, then

$\big{\\{}w\cap
u:w\in\operatorname{Set}_{\varphi,\overline{\psi}}(\overline{a}_{2})\big{\\}}=\big{\\{}w\cap
u:w\in\operatorname{Set}_{\varphi,\overline{\psi}}(\overline{a}_{3})\big{\\}}.$

###### Proof.

Let $\overline{b}_{0}\in\varphi(M,\overline{a}_{0})$ and
$\overline{b}_{1}\in\varphi(M,\overline{a}_{1})$ be such that
$u_{0}=\mathrm{set}_{\overline{\psi}}(\overline{b}_{0},\overline{a}_{0})$ and
$u_{1}=\mathrm{set}_{\overline{\psi}}(\overline{b}_{1},\overline{a}_{1})$.
Suppose that
$w\in\operatorname{Set}_{\varphi,\overline{\psi}}(\overline{a}_{2})$. Then for
some $\overline{b}_{2}\in\varphi(M,\overline{a}_{2})$ we have
$w=\mathrm{set}_{\overline{\psi}}(\overline{b}_{2},\overline{a}_{2})$. We have
to find
$w^{\prime}\in\operatorname{Set}_{\varphi,\overline{\psi}}(\overline{a}_{3})$
such that $w^{\prime}\cap u=w\cap u.$

Set $\overline{b}_{3}:=\overline{b}_{0}-\overline{b}_{1}+\overline{b}_{2}$.
Note that

$l=0,1,2\implies~{}\overline{b}_{l}^{\frown}\overline{a}_{l}\in\varphi(M),$

hence, as $\varphi\in\mathrm{Affine}_{1},$ we have

$\overline{b}_{0}^{\frown}\overline{a}_{0}-\overline{b}_{1}^{\frown}\overline{a}_{1}+\overline{b}_{2}^{\frown}\overline{a}_{2}\in\varphi(M).$

Clearly,

$\overline{b}_{3}^{\frown}\overline{a}_{3}=\overline{b}_{0}^{\frown}\overline{a}_{0}-\overline{b}_{1}^{\frown}\overline{a}_{1}+\overline{b}_{2}^{\frown}\overline{a}_{2}\in\varphi(M).$

According to its definition, $\overline{b}_{3}\in\varphi(M,\overline{a}_{3}).$

Let
$w^{\prime}=\mathrm{set}_{\overline{\psi}}(\overline{b}_{3},\overline{a}_{3})$.
We show that $w^{\prime}\cap u=w\cap u.$ Suppose $\alpha\in u.$ Then, we have
$\alpha\in u_{0}\cap u_{1}$, and hence
$\overline{b}_{j}^{\frown}\overline{a}_{j}\in\psi_{\alpha}(M)$, for $j=0,1$.
Thus as $\psi_{\alpha}\in\mathrm{Affine}_{1}$, we have

$\begin{array}[]{ll}\alpha\in
w^{\prime}&\iff\overline{b}_{3}^{\frown}\overline{a}_{3}\in\psi_{\alpha}(M)\\\
&\iff\overline{b}_{2}^{\frown}\overline{a}_{2}\in\psi_{\alpha}(M)\\\
&\iff\alpha\in w.\end{array}$

Suppose
$w^{\prime}\in\operatorname{Set}_{\varphi,\overline{\psi}}(\overline{a}_{3}).$
By symmetry, $w\cap u=w^{\prime}\cap u$ for some
$w\in\operatorname{Set}_{\varphi,\overline{\psi}}(\overline{a}_{2}).$ The
claim follows. ∎

Let us apply the previous claim and observe that:

######  Claim 2.24.

Let $\varepsilon=\operatorname{lg}(\overline{y})<\theta$ and
$\overline{a}_{l}\in{}^{\varepsilon}M$ for $l=0,1,2$, and set
$\overline{a}_{3}:=\overline{a}_{0}-\overline{a}_{1}+\overline{a}_{2}$. If
$\Upsilon\subseteq\bigcap_{l\leq
2}\operatorname{inter}_{\varphi,\overline{\psi}}(\overline{a}_{l})$, then
$\Upsilon\subseteq{\operatorname{inter}_{\varphi,\overline{\psi}}(\overline{a}_{3})}.$

###### Proof.

Let $(w_{0},w_{1})\in\Upsilon$. We shall prove that
$(w_{0},w_{1})\in\operatorname{inter}(\overline{a}_{3}).$ For $j\leq 2,$ as
$(w_{0},w_{1})\in\Upsilon\subseteq\operatorname{inter}_{\varphi,\overline{\psi}}(\overline{a}_{j})$,
there is a pair
$u_{j,0},u_{j,1}\in\mathrm{set}_{\varphi,\overline{\psi}}(\overline{a}_{j})$
witnessing it. Namely, we have

$w_{1}\subseteq u_{j,1}\text{~{}and~{}}u_{j,0}\cap w_{1}=w_{0}.$

Now, we can find $\overline{b}_{j,0},\overline{b}_{j,1}$ such that
$\rm{set}(\overline{b}_{j,0},\overline{a}_{j})=u_{j,0}$ and
$\rm{set}(\overline{b}_{j,1},\overline{a}_{j})=u_{j,1}.$ Set

  1. $\bullet$

$\overline{b}_{3,0}:=\overline{b}_{0,0}-\overline{b}_{1,0}+\overline{b}_{2,0}$,

  2. $\bullet$

$\overline{b}_{3,1}:=\overline{b}_{0,1}-\overline{b}_{1,1}+\overline{b}_{2,1}.$

In the light of Claim 2.23, there exist
$u_{3,1}\in\mathrm{set}_{\varphi,\overline{\psi}}(\overline{b}_{3})$ and
$u_{3,0}\in\mathrm{set}_{\varphi,\overline{\psi}}(\overline{b}_{3})$ such that
the following two equalities are valid:

  1. (A)

$u_{3,1}\cap(u_{0,1}\cap u_{1,1})=u_{2,1}\cap(u_{0,1}\cap u_{1,1})$, and

  2. (B)

$u_{3,0}\cap(u_{0,0}\cap u_{1,0})=u_{2,0}\cap(u_{0,0}\cap u_{1,0})$.

By clause (1), we have $u_{3,1}\supseteq w_{1}$. Hence

$\begin{array}[]{ll}w_{0}&\subseteq u_{3,1}\cap w_{1}\\\
&=u_{3,1}\cap(u_{0,1}\cap u_{1,1})\cap w_{1}\\\
&=\big{(}u_{3,1}\cap(u_{0,1}\cap u_{1,1})\big{)}\cap(u_{0,1}\cap u_{1,1})\cap
w_{1}\\\ &\stackrel{{\scriptstyle(2)}}{{=}}\big{(}u_{2,1}\cap(u_{0,1}\cap
u_{1,1})\big{)}\cap(u_{0,1}\cap u_{1,1})\cap w_{1}\\\ &\subseteq
w_{0},\end{array}$

and so

$u_{3,1}\cap w_{1}=w_{0}.$

The claim follows. ∎

Now, we are ready to complete the proof of Proposition 2.22. To this end, we
fix the following data:

  1. $\bullet_{1}$

$\vartheta(\overline{y})=\theta_{\varphi,\overline{\psi},\Upsilon}(\overline{y}),$

  2. $\bullet_{2}$

$\overline{a}_{0},\overline{a}_{1},\overline{a}_{2}\in\vartheta(M)$,

  3. $\bullet_{3}$

$\overline{a}_{3}:=\overline{a}_{0}-\overline{a}_{1}+\overline{a}_{2}$.

This gives us
$\Upsilon\subseteq\operatorname{inter}_{\varphi,\overline{\psi}}(\overline{a}_{l})$
for $l\leq 2$. Thanks to Claim 2.24, we know
$\Upsilon\subseteq{\operatorname{inter}_{\varphi,\overline{\psi}}(\overline{a}_{3})}.$
According to Definition 2.21(c) one has $\overline{a}_{3}\in\vartheta(M)$.
Consequently, $\vartheta(\overline{y})\in\mathrm{Affine}_{1}$, and the
proposition follows. ∎

##  3\. Additive frames

In this section we introduce the concept of an additive frame. Each additive
frame contains, among other things, an abelain group. We will show that each
abelian group can be realized in this way. In particular, the main result of
this section is Theorem 3.14.

The following is one of our main and new frameworks:

######  Definition 3.1.

  1. (A)

We say

$\mathbf{f}:=(M_{\mathbf{f}},\mathscr{L}_{\mathbf{f}},\lambda_{\mathbf{f}},\kappa_{\mathbf{f}},\theta_{\mathbf{f}},\Omega_{\mathbf{f}})=(M,\mathscr{L},\lambda,\kappa,\theta,\Omega)$

is a _general frame_ if:

    1. (a)

$M$ is a $\tau_{M}$-model.

    2. (b)

$\mathscr{L}$ is a class or set of formulas in the vocabulary $\tau_{M},$ such
that each $\varphi\in\mathscr{L}$ has the form $\varphi(\overline{x}),\
\overline{x}$ of length $<\theta$.

    3. (c)

For every $\overline{a}\in{}^{\varepsilon}M,~{}\varepsilon<\theta$, there is a
formula $\varphi_{\overline{a}}(\overline{x})\in\mathscr{L}$ such that:

      1. $\bullet_{1}$

$\overline{a}\in\varphi_{\overline{a}}(M),$

      2. $\bullet_{2}$

(the minimality condition) if $\psi(\overline{x})\in\mathscr{L}$ and
$\overline{a}\in\psi(M),$ then $\varphi_{\overline{a}}(M)\subseteq\psi(M).$

    4. (d)

      1. $\bullet_{1}$

if $\varphi_{\alpha}(\overline{x})\in\mathscr{L}$ for $\alpha<\kappa$, then
for some $\alpha<\beta<\kappa$, we have
$\varphi_{\alpha}(M)\supseteq\varphi_{\beta}(M),$

      2. $\bullet_{2}$

if $\varphi_{\alpha,\beta}(\overline{x},\overline{y})\in\mathscr{L}$ for
$\alpha<\beta<\lambda$, then for some
$\alpha_{1}<\alpha_{2}<\alpha_{3}<\lambda$ we have
$\varphi_{\alpha_{1},\alpha_{2}}(M)\supseteq\varphi_{\alpha_{1},\alpha_{3}}(M),\varphi_{\alpha_{2},\alpha_{3}}(M)$.

    5. (e)

$\lambda,\kappa$ are regular and
$\lambda\geq\kappa\geq\theta\geq|\Omega|+|\tau_{M}|,$ where $\Omega$ is a set
of cardinals such that $1\in\Omega$ and all other cardinals in it are
infinite.

  2. (B)

We say a general frame $\mathbf{f}$ is an _additive frame_ if in addition, it
satisfies:

    1. (6)

$(|M|,+^{M},-^{M},0^{M})$ is an abelian group. Moreover, $M$ is an additive
$\theta$-model.

    2. (7)

If $\varphi(\overline{x}_{u})\in\mathscr{L}$, then
$\varphi(M)\subseteq{}^{u}M$ is a sub-group of $M$.

  3. (C)

An additive frame $\mathbf{f}$ is an _additive + frame_ if $M_{\mathbf{f}}$
has cardinality greater or equal to $\lambda$.

######  Remark 3.2.

Given a general frame $\mathbf{f}$ as above, we always assume that the
language $\mathscr{L}$ is closed under permutation of variables, adding dummy
variables and finite conjunction.

The next lemma is a criterion for an additive frame to be additive+.

######  Lemma 3.3.

Suppose $\mathbf{f}=(M,\mathscr{L},\lambda,\kappa,\theta,\Omega)$ is an
additive frame. Then it is an additive+ frame if and only if for each
$\varepsilon\in(0,\theta)$, there exists some $\bar{a}\in$${}^{\varepsilon}M$
such that $\varphi_{\bar{a}}$ has cardinality $\geq\lambda$.

###### Proof.

he assumption clearly implies $\mathbf{f}$ is additive+. To see the other
direction, suppose $\mathbf{f}$ is an additive+ frame and
$\varepsilon\in(0,\theta).$ Suppose by the way of contradiction,
$|\varphi_{\bar{a}}|<\lambda$ for all $\bar{a}\in$${}^{\varepsilon}M$. By
induction on $\alpha<\kappa$ we can find a sequence
$\langle\bar{a}_{\beta}:\beta<\kappa\rangle$ such that for each
$\beta<\kappa,$
$\bar{a}_{\beta}\notin\bigcup_{\alpha<\beta}\varphi_{\bar{a}_{\alpha}}$. This
contradicts Definition 3.1(4)(a). ∎

The following defines a partial order relation on formulas of a frame.

######  Definition 3.4.

Assume $\mathbf{f}$ is a general frame, and let
$\psi(\overline{x}),\varphi(\overline{x})$ be in $\mathscr{L}_{\mathbf{f}}$.

  1. (A)

We say $\psi(\overline{x})\leq\varphi(\overline{x})$ if
$\varphi(M)\supseteq\psi(M).$

  2. (B)

We say $\psi(\overline{x})$ and $\varphi(\overline{x})$ are equivalent,
denoted by $\psi(\overline{x})\equiv\varphi(\overline{x})$, if
$\varphi(M)=\psi(M).$

  3. (C)

Suppose $\overline{a},\overline{b}\in{}^{\varepsilon}M$. We say
$\overline{a}\leq\overline{b}$ (resp. $\overline{a}\equiv\overline{b}$) if
$\varphi_{\overline{a}}\leq\varphi_{\overline{b}}$ (resp.
$\varphi_{\overline{a}}\equiv\varphi_{\overline{b}}$).

######  Notation 3.5.

Assume $\mathbf{f}=(M,\lambda,\kappa,\theta,\Omega)$ is an additive frame. Let
$\overline{a}_{l}=\langle
a_{l,\zeta}:\zeta<\varepsilon\rangle\in{}^{\varepsilon}M$ for $l<n$. We set:

  * •

$-\overline{a}_{l}:=\langle-a_{l,\zeta}:\zeta<\varepsilon\rangle,$

  * •

$\overline{a}_{1}+\overline{a}_{2}:=\langle
a_{1,\zeta}+a_{2,\zeta}:\zeta<\varepsilon\rangle,$

  * •

$\sum_{l<n}\overline{a}_{l}:=\langle\sum_{l<n}a_{l,\zeta}:\zeta<\varepsilon\rangle,$

  * •

$\overline{a}-\overline{b}:=\overline{a}+(-\overline{b}).$

######  Lemma 3.6.

Suppose $\mathbf{f}=(M,\lambda,\kappa,\theta,\Omega)$ is an additive frame,
$\overline{a}\in{}^{\varepsilon}M,\varphi=\varphi_{\overline{a}}$ and
$\overline{a}_{\alpha}\in\varphi(M)$ for $\alpha<\lambda$. Then for some
$\overline{\beta},\overline{\gamma}$ we have:

  1. (A)

$\overline{\beta}=\langle\beta_{i}:i<\lambda\rangle\in{}^{\lambda}\lambda$ is
increasing,

  2. (B)

$\overline{\gamma}=\langle\gamma_{i}:i<\lambda\rangle\in{}^{\lambda}\lambda$
is increasing,

  3. (C)

$\beta_{i}<\gamma_{i}<\beta_{i+1},$ for all $i<\lambda$,

  4. (D)

$\overline{a}-\overline{a}_{\beta_{i}}+\overline{a}_{\gamma_{i}}$ is
equivalent to $\overline{a},$ for all $i<\lambda.$

###### Proof.

First, we reduce the lemma to the following claim:

  1. $(*)$

It is suffice to prove, for each sequences
$\overline{a},\langle\overline{a}_{\alpha}:\alpha<\lambda\rangle$ as above,
there are $\beta<\gamma<\lambda$ such that
$\overline{a}-\overline{a}_{\beta}+\overline{a}_{\gamma}$ is equivalent to
$\overline{a}.$

To see this, suppose $(*)$ holds. By induction on $i<\lambda,$ we define the
increasing sequences $\langle\beta_{i}:i<\lambda\rangle$ and
$\langle\gamma_{i}:i<\lambda\rangle$ as requested. Thus suppose that
$i<\lambda$, and we have defined $\langle\gamma_{j},\beta_{j}:j<i\rangle$. In
order to define $(\beta_{i},\gamma_{i})$, we let

$\alpha_{*}:=\sup\\{\gamma_{j}+\beta_{j}+1:j<i\\}.$

Since $\lambda$ is regular, $\alpha_{*}<\lambda$. Now, apply $(\ast)$ to
$\overline{a}$ and
$\langle\overline{a}_{\alpha_{*}+\alpha}:\alpha<\kappa\rangle$. This gives us
$\beta<\gamma<\kappa$ such that

$\overline{a}-\overline{a}_{\alpha_{*}+\beta}+\overline{a}_{\alpha_{*}+\gamma}\equiv\overline{a}.$

Thus it suffices to set $\beta_{i}=\alpha_{*}+\beta$ and
$\gamma_{i}=\alpha_{*}+\gamma.$

So, things are reduced in showing $(\ast)$ holds. To see this, we define the
formula $\varphi_{\beta\gamma}$ as

$(+)\quad\quad\quad\varphi_{\beta\gamma}:=\varphi_{\overline{a}-\overline{a}_{\beta}+\overline{a}_{\gamma}},$

where $\beta<\gamma<\lambda$. Note that
$\overline{a},\overline{a}_{\beta},\overline{a}_{\gamma}\in\varphi_{\overline{a}}(M),$
hence as $\varphi_{\overline{a}}(M)$ is a submodel,
$\overline{a}-\overline{a}_{\beta}+\overline{a}_{\gamma}\in\varphi_{\overline{a}}(M).$
Thanks to the minimality condition from Definition 3.1(3)(b), this implies
that

$\varphi_{\overline{a}}\geq\varphi_{\beta,\gamma}.$

Thus, it is sufficient to find $\beta<\gamma<\kappa$ such that
$\varphi_{\overline{a}}\leq\varphi_{\beta,\gamma}$. By the property presented
in Definition 3.1(4)(b) there are $\alpha_{1}<\alpha_{2}<\alpha_{3}<\kappa$
such that

$\varphi_{\alpha_{1},\alpha_{2}}\geq\varphi_{\alpha_{1},\alpha_{3}},\varphi_{\alpha_{2},\alpha_{3}}.$

So,

  1. (A)

$\overline{a}-\overline{a}_{\alpha_{1}}+\overline{a}_{\alpha_{2}}\in\varphi_{\alpha_{1},\alpha_{2}}(M)$,

  2. (B)

$\overline{a}-\overline{a}_{\alpha_{1}}+\overline{a}_{\alpha_{3}}\in\varphi_{\alpha_{1},\alpha_{3}}(M)\subseteq\varphi_{\alpha_{1},\alpha_{2}}(M)$,

  3. (C)

$\overline{a}-\overline{a}_{\alpha_{2}}+\overline{a}_{\alpha_{3}}\in\varphi_{\alpha_{2},\alpha_{3}}(M)\subseteq\varphi_{\alpha_{1},\alpha_{2}}(M).$

Hence

$\begin{array}[]{clcr}\overline{a}&=(1)-(2)+(3)\in\varphi_{\alpha_{1},\alpha_{2}}(M).\end{array}$

Combining this with the minimality property from Definition 3.1(3)(b) we
observe that $\varphi_{\overline{a}}\leq\varphi_{\alpha_{1},\alpha_{2}}$. Thus
it suffices to take $\beta=\alpha_{1}$ and $\gamma=\alpha_{2}$. ∎

######  Corollary 3.7.

Suppose $\mathbf{f}=(M,\lambda,\kappa,\theta,\Omega)$ is an additive frame.
The following assertions are valid:

  1. (A)

Suppose $\varphi_{\overline{a}}(M)$ has cardinality $\geq\lambda$. Then there
is $\overline{b}\in\varphi_{\overline{a}}(M)$ such that
$\overline{b}\neq\overline{a}$ and $\overline{b}$ is equivalent to
$\overline{a}.$

  2. (B)

If for some $\overline{c}\in$${}^{\varepsilon}M$, $\varphi_{\overline{c}}(M)$
has cardinality $\geq\lambda,$ then for all
$\overline{a}\in$${}^{\varepsilon}M$ the set

$\\{\overline{b}\in\varphi_{\overline{a}}(M):\overline{b}\text{~{}is
equivalent to~{}}\overline{a}\\}$

has cardinality $\geq\lambda.$

  3. (C)

If $\mathbf{f}$ ia an additive+ frame and $\varepsilon\in(0,\theta)$, then for
all $\overline{a}\in$${}^{\varepsilon}M$, the set
$\\{\overline{b}\in\varphi_{\overline{a}}(M):\overline{b}\text{~{}is
equivalent to~{}}\overline{a}\\}$ has cardinality $\geq\lambda.$

###### Proof.

(1) Since $\varphi_{\overline{a}}(M)$ has cardinality $\geq\lambda$, we can
take a sequence $\langle\overline{a}_{\alpha}:\alpha<\lambda\rangle$ of length
$\lambda$ of pairwise distinct elements of $\varphi_{\overline{a}}(M)$ with no
repetition. We apply Lemma 3.6 to find increasing sequences
$\overline{\beta}=\langle\beta_{i}:i<\lambda\rangle$ and
$\overline{\gamma}=\langle\gamma_{i}:i<\lambda\rangle$ such that for all
$i<\lambda$

  1. (A)

$\beta_{i}<\gamma_{i},$

  2. (B)

$\overline{a}-\overline{a}_{\beta_{i}}+\overline{a}_{\gamma_{i}}$ is
equivalent to $\overline{a}.$

Set
$\overline{b}_{i}:=\overline{a}-\overline{a}_{\beta_{i}}+\overline{a}_{\gamma_{i}}$.
Since $\overline{a}_{\alpha}$’s are distinct, we deduce that
$\overline{a}\neq\overline{b}_{i}$, for at least one $i<\lambda$. Thanks to
(b), we know $\overline{b}_{i}$ is equivalent to $\overline{a}.$

(2) Let $X=\\{\overline{b}\in\varphi_{\overline{a}}(M):\overline{b}\text{~{}is
equivalent to~{}}\overline{a}\\},$ and let $\mu=|X|.$ Suppose towards
contradiction that $\mu<\lambda,$ and let
$\langle\overline{b}_{i}:i<\mu\rangle$ be enumerates $X$. Let also
$\langle\overline{c}_{\alpha}:\alpha<\lambda\rangle$ be a sequence of length
$\lambda$ of pairwise distinct elements of $\varphi_{\overline{c}}(M)$ with no
repetition. By induction on $\alpha<\lambda$ we can find
$\xi_{\alpha}<\lambda$ such that

  1. $(*)_{\alpha}$:

$\overline{a}_{\xi_{\alpha}}\notin\\{\overline{b}_{i}-\overline{c}+\overline{c}_{\xi_{\beta}}:\beta<\alpha,i<\mu\\}$.

By the argument of clause (1), applied to the sequence
$\langle\overline{c}_{\xi_{\alpha}}:\alpha<\lambda\rangle$, we can find some
$\beta_{\iota}<\gamma_{\iota}<\lambda$ such that
$\overline{a}-\overline{c}_{\xi_{\beta_{\iota}}}+\overline{c}_{\xi_{\gamma_{\iota}}}$
is equivalent (but not equal) to $\overline{c}.$ Thus for some
$i<\mu,\overline{a}-\overline{c}_{\xi_{\beta_{\iota}}}+\overline{c}_{\xi_{\gamma_{\iota}}}=\overline{b}_{i}$.
But then

$\overline{c}_{\xi_{\gamma_{\iota}}}=\overline{b}_{i}-\overline{a}+\overline{c}_{\xi_{\beta_{\iota}}},$

which contradicts $(*)_{\gamma_{\iota}}$.

(3) By Lemma 3.3 and clause (2). ∎

######  Lemma 3.8.

  1. (A)

Suppose $\mathbf{f}=(M,\lambda,\kappa,\theta,\Omega)$ is a general frame,
$\varepsilon<\theta$ and $\overline{a}_{\alpha}\in{}^{\varepsilon}M$ for
$\alpha<\kappa$. Then there is some $\alpha<\kappa$ such that the set
$\\{\beta<\kappa:\overline{a}_{\beta}\in\varphi_{\overline{a}_{a}}(M)\\}$ is
unbounded in $\kappa.$

  2. (B)

In clause (1), we can replace $\kappa$ by any cardinal
$\kappa^{\prime}\geq\kappa.$

###### Proof.

We prove clause (2). Suppose on the way of contradiction that, for each
$\alpha<\kappa^{\prime},$ the set
$X_{\alpha}=\\{\beta<\kappa^{\prime}:\overline{a}_{\beta}\in\varphi_{\overline{a}_{a}}(M)\\}$
is bounded in $\kappa^{\prime}.$ So,

  1. $(*)_{1}$

$\forall\alpha<\kappa^{\prime},\exists\alpha<\beta_{\alpha}<\kappa^{\prime}$
such that $\forall\beta\geq\beta_{\alpha}$ we have
$\varphi_{\alpha}\ngeq\varphi_{\beta}.$

We define an increasing and continuous sequence
$\langle\zeta_{\alpha}:\alpha<\kappa\rangle$ of ordinals less than
$\kappa^{\prime}$, by induction on $\alpha$ as follows:

  * •

$\zeta_{0}:=0,$

  * •

$\zeta_{\alpha+1}:=\beta_{\zeta_{\alpha}}$,

  * •

$\zeta_{\delta}:=\lim_{\alpha<\delta}\zeta_{\alpha}$ for limit ordinal
$\delta$.

Consider the sequence $\\{\varphi_{\zeta_{\alpha}}:\alpha<\kappa\\}$, and
apply the property presented in Definition 3.1(4)(a) to find
$\gamma<\delta<\kappa$ such that

  1. $(*)_{2}$

$\varphi_{\zeta_{\gamma}}\geq\varphi_{\zeta_{\delta}}.$

Since $\gamma<\delta$,
$\zeta_{\delta}\geq\zeta_{\gamma+1}=\beta_{\zeta_{\gamma}}$. We apply
$(*)_{1}$ for $\alpha:=\zeta_{\gamma}$ and
$\beta:=\zeta_{\delta}\geq\beta_{\alpha}$. This gives us
$\varphi_{\zeta_{\gamma}}\ngeq\varphi_{\zeta_{\delta}},$ which contradicts
$(*)_{2}$. ∎

In what follows we need to use a couple of results from [29]. To make the
paper more self contained, we borrow some definitions and results from it.

######  Definition 3.9.

  1. (A)

By a  tree we mean a partially-ordered set $(\mathscr{T},\leq)$ such that for
all $t\in\mathscr{T}$,

$pred(t):=\\{s\in\mathscr{T}:s<t\\},$

is a well-ordered set; moreover, there is only one element $r$ of
$\mathscr{T}$, called the root of $\mathscr{T}$, such that $pred(r)$ is empty.

  2. (B)

The order-type of $pred(t)$ is called the height of $t$, denoted by ht$(t)$.

  3. (C)

The height of $\mathscr{T}$ is $\sup\\{$ht$(t)+1:t\in\mathscr{T}\\}$.

######  Definition 3.10.

  1. (A)

A quasi-order $\mathcal{Q}$ is a pair $(\mathcal{Q},\leq_{\mathcal{Q}})$ where
$\leq_{\mathcal{Q}}$ is a reflexive and transitive binary relation on
$\mathcal{Q}$.

  2. (B)

$\mathcal{Q}$ is called $\kappa$–narrow, if there is no antichain in
$\mathcal{Q}$ of size $\kappa$, i.e., for every
$f:\kappa\rightarrow\mathcal{Q}$ there exist $\nu\neq\mu$ such that
$f(\nu)\leq_{\mathcal{Q}}f(\mu)$.

  3. (C)

For a quasi-order $\mathcal{Q}$, a $\mathcal{Q}$-labeled tree is a pair
$(\mathscr{T},\Phi_{\mathscr{T}})$ consisting of a tree $\mathscr{T}$ of
height $\leq\omega$ and a function
$\Phi_{\mathscr{T}}:\mathscr{T}\rightarrow\mathcal{Q}$.

  4. (D)

$\mathcal{Q}$ is $\kappa$-well ordered if for every sequence $\left\langle
q_{i}:i<\kappa\right\rangle$ of elements of $\mathcal{Q},$ there are
$i<j<\kappa$ such that $q_{i}\leq_{\mathcal{Q}}q_{j}.$

######  Remark 3.11.

On any set of $\mathcal{Q}$-labeled trees we define a quasi-order by:
$(\mathscr{T}_{1},\Phi_{1})\preceq(\mathscr{T}_{2},\Phi_{2})$ if and only if
there is a function $\nu:\mathscr{T}_{1}\rightarrow\mathscr{T}_{2}$ equips
with the following properties:

  * a)

for all $t\in\mathscr{T}_{1}$,
$\Phi_{1}(t)\leq_{\mathcal{Q}}\Phi_{2}(\nu(t))$,

  * b)

$t\leq_{\mathscr{T}_{1}}t^{\prime}\Longrightarrow\nu(t)\leq_{\mathscr{T}_{2}}\nu(t^{\prime}),$

  * c)

for all $t\in\mathscr{T}_{1}$,
ht${}_{\mathscr{T}_{1}}(t)=$ht${}_{\mathscr{T}_{2}}(\nu(t))$.

######  Definition 3.12.

  1. (A)

Given infinite cardinals $\kappa$ and $\mu,$ the notation
$\kappa\longrightarrow(\omega)_{\mu}^{<\omega}$ means that: for every function
$f:[\kappa]^{<\omega}\to\mu$, there exists an infinite subset
$X\subseteq\kappa$ and a function $g:\omega\to\mu$ such that $f(Y)=g(|Y|)$ for
all finite subsets $Y$ of $X$.

  2. (B)

Let $\kappa_{\rm{beau}}$ denote the first beautiful111This also is called the
first $\omega$-Erdös cardinal. cardinal. This is defined as the smallest
cardinal $\kappa$ such that $\kappa\longrightarrow(\omega)_{2}^{<\omega}$.

  3. (C)

Given a ring $R$ and an infinite cardinal $\theta$, let
$\kappa_{\rm{beau}}(R,\theta)$ denote the least cardinal $\kappa$ such that
$\kappa\longrightarrow(\omega)_{|R|+\theta^{<\theta}}^{<\omega}$.

  4. (D)

Given a vocabulary $\tau$, let $\kappa_{\rm{beau}}(\tau,\theta)$ denote the
least cardinal $\kappa$ such that
$\kappa\longrightarrow(\omega)_{|\tau|+\theta^{<\theta}}^{<\omega}$. If
$\theta=\aleph_{0}$, we may omit it.

Now, we can state:

######  Fact 3.13.

(Shelah, [29, theorems 5.3+ 2.10]) Let $\mathcal{Q}$ be a quasi-order of
cardinality $<\kappa_{\rm{beau}}$, and $\mathcal{S}$ be a set of
$\mathcal{Q}$-labeled trees with $\leq\omega$ level. Then $\mathcal{S}$ is
$\kappa_{\rm{beau}}$-narrow, and even $\kappa$-well ordered.

We are now ready to state and prove the main result of this section.

######  Theorem 3.14.

  1. (i)

Assume $M$ is an $R$-module and $\kappa=\kappa_{\rm{beau}}(R,\theta)$ and
$\Omega$ is such that $1\in\Omega$ and $|\Omega|\leq\theta$. Also assume that
$\lambda\geq\kappa$ is regular and satisfies $\lambda\to(\kappa+1)_{4}^{3}$.
The following hold:

    1. (a)

$\mathbf{f}=(M,\lambda,\kappa,\theta,\Omega)$ is an additive frame, whenever

$\mathscr{L}\subseteq\\{\varphi(\overline{x}):\varphi\in\mathscr{L}_{\infty,\theta}^{\rm{cop}}(\tau_{M}),\rm{lg}(\overline{x})<\theta\\}$

is closed under arbitrary conjunctions.

    2. (b)

$\mathbf{f}=(M,\lambda,\kappa,\theta,\Omega)$ is an additive frame, whenever

$\mathscr{L}\subseteq\\{\varphi(\overline{x}):\varphi\in\mathscr{L}_{\infty,\theta}^{\rm{co}}(\tau_{M}),\rm{lg}(\overline{x})<\theta\\}$

is closed under arbitrary conjunctions.

  2. (ii)

If $M$ is a $\tau$-model, $\kappa=\kappa_{\rm{beau}}(\tau,\theta)$ and
$\lambda\geq\kappa$ is regular and satisfies $\lambda\to(\kappa+1)_{4}^{3}$.
Then $\mathbf{f}=(M,\lambda,\kappa,\theta,\Omega)$ is a additive frame,
whenever

$\mathscr{L}\subseteq\\{\varphi(\overline{x}):\varphi\in\mathscr{L}_{\infty,\theta}^{\rm{co}}(\tau_{M}),\rm{lg}(\overline{x})<\theta\\}$

is closed under arbitrary conjunctions.

  3. (iii)

Furthermore, if $|M|\geq\lambda$, then in items (i) and (ii), $\mathbf{f}$
becomes an additive+ frame.

###### Proof.

(i): We have to show that $\mathbf{f}$ satisfies the relevant items of
Definition 3.1. Items (1), (2) and (5) are true by definition. As $M$ is an
$R$-module, clause (6) is valid. The validity of (7) follows from Lemma 2.15.

Let us consider clause (3). Thus suppose that
$\overline{a}\in{}^{\varepsilon}M$, where $\varepsilon<\theta$. We are going
to find some $\varphi\in\mathscr{L}$ such that:

  1. $\bullet$

$\overline{a}\in\varphi(M),$

  2. $\bullet$

if $\psi\in\mathscr{L}$ and $\overline{a}\in\psi(M),$ then
$\varphi(M)\subseteq\psi(M).$

For any $\overline{b}\in{}^{\varepsilon}M,$ if there is some formula
$\varphi(\overline{x})$ such that

$(\dagger)_{1}\quad\quad\quad
M\models\varphi(\overline{a})\wedge\neg\varphi(\overline{b}),$

then let $\varphi_{\overline{b}}\in\mathscr{L}$ be such a formula. Otherwise,
let $\varphi_{\overline{b}}\in\mathscr{L}$ be any true formula such that
$\varphi_{\overline{b}}(M)={}^{\varepsilon}M$. Finally set

$(\dagger)_{2}\quad\quad\quad\varphi:=\bigwedge\\{\varphi_{\overline{b}}:\overline{b}\in{}^{\varepsilon}M\\}.$

Now, we claim that $\varphi$ is as desired. First, we check (3)(a), that is
$\overline{a}\in\varphi(M).$ As

$\varphi(M)=\bigcap_{\overline{b}\in{}^{\varepsilon}M}\varphi_{\overline{b}}(M),$

it suffices to show that $\overline{a}\in\varphi_{\overline{b}}(M)$, for
$\overline{b}\in{}^{\varepsilon}M$. Fix $\overline{b}$ as above. If there is
no formula $\varphi(\overline{x})$ as in $(\dagger)$, then
$\overline{a}\in{}^{\varepsilon}M=\varphi_{\overline{b}}(M)$, and we are done.
Otherwise, by its definition, $M\models\varphi_{\overline{b}}(\overline{a})$,
and hence, again we have $\overline{a}\in\varphi_{\overline{b}}(M)$.

To see (3)(b) holds, let $\psi(\overline{x})$ be such that
$\overline{a}\in\psi(M).$ We have to show that $\varphi(M)\subseteq\psi(M).$
Suppose by the way of contradiction that $\varphi(M)\nsubseteq\psi(M).$ Take
$\overline{b}\in\varphi(M)\setminus\psi(M)$. Now,
$M\models\neg\psi(\overline{b}),$ and by our assumption
$M\models\psi(\overline{a}).$ In particular, by our construction, the formula
$\varphi_{\overline{b}}$ satisfies

$(\dagger)_{3}\quad\quad\quad
M\models\varphi_{\overline{b}}(\overline{a})\wedge\neg\varphi_{\overline{b}}(\overline{b}).$

Now,

  1. $\bullet_{1}$

$\overline{b}\in\varphi(M)\stackrel{{\scriptstyle(+)}}{{\subseteq}}\varphi_{\overline{b}}(M)$,

  2. $\bullet_{2}$

by $(\dagger)_{3}$, $M\models\neg\varphi_{\overline{b}}(\overline{b})$, and
hence $\overline{b}\notin\varphi_{\overline{b}}(M)$.

By $\bullet_{1}$ and $\bullet_{2}$ we get a contradiction.

Now we turn to clause (4). First let us consider (4)(a). Thus suppose that
$\varphi_{\alpha}(\overline{x})\in\mathscr{L}$, for $\alpha<\kappa$, are
given. We should find some $\alpha<\beta<\kappa$ such that
$\varphi_{\alpha}\geq\varphi_{\beta}$, i.e.,
$\varphi_{\alpha}(M)\supseteq\varphi_{\beta}(M).$ To this end, first note that
we can restrict ourselves to those formulas such that both free and bounded
variables appearing in them are among $\\{x_{i}:i<\theta\\}$, the set of free
variables of $\varphi$ has the form $\\{x_{\zeta}:\zeta<\varepsilon\\}$, and
the quantifiers have the form
$\exists\bar{x}_{[\varepsilon_{0},\varepsilon_{1})}\text{~{}and~{}}\forall\bar{x}_{[\varepsilon_{0},\varepsilon_{1})},$
where $\varepsilon_{0}<\varepsilon_{1}<\theta,$ and
$\bar{x}_{[\varepsilon_{0},\varepsilon_{1})}=\langle
x_{\xi}:\varepsilon_{0}\leq\xi<\varepsilon_{1}\rangle$. In what follows,
writing a formula as $\varphi(\bar{x})$, we mean $\bar{x}$ lists the free
variables appearing in $\varphi$ in increasing order.

We can consider a formula $\varphi(\bar{x})$ as a type $(\mathscr{T},\boldmath
c)$ such that

  1. (A)

$\mathscr{T}$ is a tree with $\leq\omega$ levels with no infinite branches,

  2. (B)

$\boldmath c$ is a function with domain $\mathscr{T}$,

  3. (C)

if $t\in\mathscr{T}\setminus\max(\mathscr{T})$, then $c(t)$ is in the
following set

$\bigg{\\{}\wedge,\exists\bar{x},\forall\bar{x}:\bar{x}\emph{ has form
}\bar{x}_{[\varepsilon_{0},\varepsilon_{1})}\emph{ for some
}\varepsilon_{0}<\varepsilon_{1}<\theta\bigg{\\}},$

  4. (D)

if $t\in\max(\mathscr{T})$ then $\boldmath c(t)$ is an atomic formula in
$\tau_{M}$.

Clearly, $|\mathrm{Rang}(\boldmath c)|\leq\theta+|R|$. For each
$\alpha<\kappa$ set $\varphi_{\alpha}(\bar{x})=(\mathscr{T}_{\alpha},\boldmath
c_{\alpha}).$

Let $\mathcal{Q}$ be the range of the function $c$. Note that it is a quasi-
order under the $\leq$ relation defined in Definition 3.4, and clearly, it has
cardinality

$|\mathcal{Q}|\leq|\tau_{M}|+\theta^{<\theta}<\kappa_{\rm{beau}}.$

In particular, by Fact 3.13, applied to the sequence

$\langle(\mathscr{T}_{\alpha},\boldmath c_{\alpha}):\alpha<\kappa\rangle.$

we get some ${\alpha}<\beta$ and a function $f$ equipped with the following
property:

  1. $(\ast)$

$f$ is a 1-1 function from $\mathscr{T}_{\alpha}$ into $\mathscr{T}_{\beta}$,
which is level, order and non-order preserving and such that

$t\in\boldmath c_{\alpha}\Rightarrow\big{(}\boldmath c_{\beta}(f(t))=\boldmath
c_{\alpha}(t)\big{)}.$

Let $\mathscr{T}_{\alpha}^{\prime}:=\mathrm{Rang}(f)$ and $\boldmath
c_{\alpha}^{\prime}:=\boldmath
c_{\beta}\upharpoonright\mathscr{T}_{\alpha}^{\prime}$. By this notation,
$\varphi_{\alpha}(\bar{x})$ can also be represented by
$(\mathscr{T}_{\alpha}^{\prime},\boldmath c_{\alpha}^{\prime})$. For any
$t\in\mathscr{T}_{\alpha}^{\prime}$ we define $\varphi_{t}^{1}$ to be the
formula represented by

$\bigg{(}\\{s\in\mathscr{T}_{\alpha}^{\prime}:t\leq_{\mathscr{T}_{\alpha}^{\prime}}s\\},\boldmath{c}_{\beta}\upharpoonright\\{s\in\mathscr{T}_{\alpha}^{\prime}:t\leq_{\mathscr{T}_{\alpha}^{\prime}}s\\}\bigg{)},$

and define $\varphi_{t}^{2}$ to be the formula represented via

$\bigg{(}\\{s\in\mathscr{T}_{\beta}:t\leq_{\mathscr{T}_{\beta}^{\prime}}s\\},\boldmath
c_{\beta}\upharpoonright\\{s\in\mathscr{T}_{\beta}:t\leq_{\mathscr{T}_{\beta}^{\prime}}s\\}\bigg{)}.$

Note that the formula $\varphi_{t}^{1}$ may have fewer free variables than
$\varphi_{t}^{2}$, but we can add the remaining variables. So, we may and do
assume that

$\varphi_{t}^{\ell}=\varphi_{t}^{\ell}(\bar{x}_{t})\quad\ell=1,2.$

Now we are going to prove that for every $t$,

$\varphi^{1}_{t}(M)\supseteq\varphi^{2}_{t}(M).$

This is done by induction on the depth of $t$ inside
$\mathscr{T}_{\alpha}^{\prime}$ which is possible, as
$\mathscr{T}_{\alpha}^{\prime}$ is well-founded. By the way we defined our
trees, it suffices to deal with the following cases:

  1. Case 1): $c_{\beta}(t)$ is a basic formal, i.e., an atomic formula or its negation.

  2. Case 2): $c_{\beta}(t)$ is $\wedge$.

  3. Case 3): $c_{\beta}(t)$ is $\exists\overline{x}^{\prime}$ or just $c_{\beta}(t)$ is $\exists^{\sigma}\overline{x}^{\prime}$.

  4. Case 4) $c_{\beta}(t)$ is $\forall\overline{x}^{\prime}$ or just $c_{\beta}(t)$ is $\forall^{\sigma}\overline{x}^{\prime}$.

Let discuss each cases separably:

Case 1): Here, $t$ is a maximal node of the tree $\mathscr{T}_{\beta}$. Hence,
necessarily a maximal node of $\mathscr{T}_{\alpha}^{\prime}$, and recall that

$\varphi^{1}_{t}=c_{\beta}(t)=\varphi^{2}_{t}.$

Consequently, the conclusion is obvious.

Case 2): Here, we have

  1. $\bullet$

$\varphi^{2}_{t}:=\bigwedge\\{\varphi^{2}_{s}:s\in\mathrm{suc}_{\mathscr{T}_{\beta}}(t)\\}$,

  2. $\bullet$

$\varphi^{1}_{t}:=\bigwedge\\{\varphi^{1}_{s}:s\in\mathrm{suc}_{\mathscr{T}_{\alpha}^{\prime}}(t)\\}$.

By the choice of the $\mathscr{T}_{\alpha}^{\prime}$ and the function $f$, we
have

$\mathrm{suc}_{\mathscr{T}_{\alpha}^{\prime}}(t)\subseteq\mathrm{suc}_{\mathscr{T}_{\beta}}(t).$

Now, due to the induction hypotheses

$s\in\mathrm{suc}_{\mathscr{T}_{\alpha}^{\prime}}(t)\Longrightarrow\varphi^{1}_{s}(M)\supseteq\varphi^{2}_{s}(M).$

According to the definition of sanctification for $\wedge$ we are done,
indeed:

$\begin{array}[]{ll}\varphi^{1}_{t}(M)&=\bigcap\\{\varphi^{1}_{s}(M):s\in\mathrm{suc}_{\mathscr{T}_{\alpha}^{\prime}}(t)\\}\\\
&\supseteq\bigcap\\{\varphi^{2}_{s}(M):s\in\mathrm{suc}_{\mathscr{T}_{\alpha}^{\prime}}(t)\\}\\\
&\supseteq\bigcap\\{\varphi^{s}_{s}(M):s\in\mathrm{suc}_{\mathscr{T}_{\beta}}(t)\\}.\end{array}$

Case 3): Let
$\overline{x}^{\prime}_{t}=\overline{x}_{s}^{\frown}\overline{x}^{\prime}$,
and recall that $\mathrm{suc}_{\mathscr{T}_{\beta}}(t)$ is singleton, and also
$\mathrm{suc}_{\mathscr{T}_{\alpha}}(f^{(-1)}(t))$ is singleton, because
$\boldmath c_{\beta}(t)=\boldmath c_{\alpha}(f^{-1}(t))$. This implies that
$\mathrm{suc}_{\mathscr{T}_{\alpha}^{\prime}}(t)$ is singleton, say
$\mathrm{suc}_{\mathscr{T}_{\alpha}^{\prime}}(t)=\\{s\\}=\mathrm{suc}_{\mathscr{T}_{\beta}}(t)$.
In order to see $\varphi^{1}_{t}(M)\supseteq\varphi^{2}_{t}(M)$, we take
$\overline{a}\in{}^{\lg(\overline{x})}M$ be such that
$M\models\varphi^{2}_{t}(\overline{a})$ and shall show that
$M\models\varphi^{1}_{t}(\overline{a})$. Indeed,

$\varphi^{\ell}_{t}(\overline{x}):=(\exists^{\sigma}\overline{x}^{\prime})\varphi^{\ell}_{s}(\overline{x}_{s},\overline{x}^{\prime})\quad\ell=1,2,$

and since

$M\models(\exists^{\sigma}\overline{x}^{\prime})\varphi^{2}_{s}(\overline{a},\overline{x}^{\prime}),$

necessarily, for some pairwise disjoint
$\overline{b}_{\zeta}\in{}^{\lg(\overline{x}^{\prime})}M$ we have

$M\models\varphi^{2}_{s}(\overline{a},\overline{b}_{\zeta}).$

Thanks to the inductive hypothesis, we know

$M\models\varphi^{1}_{s}(\overline{a},\overline{b}_{\zeta}).$

According to the definition of sanctification, we have

$M\models(\exists^{\sigma}\overline{x}^{\prime})\varphi^{1}_{s}(\overline{a},\overline{x}^{\prime}),$

which means that $M\models\varphi^{1}_{t}(\overline{a})$, as promised.

Case 4): Let
$\overline{x}^{\prime}_{t}=\overline{x}_{s}^{\frown}\overline{x}^{\prime}$,
and recall that $\mathrm{suc}_{\mathscr{T}_{\beta}}(t)$ is singleton, and also
$\mathrm{suc}_{\mathscr{T}_{\alpha}}(f^{-1}(t))$ is singleton, because
$\boldmath c_{\alpha}(f^{-1}(t))\subseteq\boldmath c_{\beta}(t)$. This implies
that $\mathrm{suc}_{\mathscr{T}_{\alpha}^{\prime}}(t)$ is singleton, say
$\mathrm{suc}_{\mathscr{T}_{\alpha}^{\prime}}(t)=\\{s\\}=\mathrm{suc}_{\mathscr{T}_{\beta}}(t)$.
In order to see $\varphi^{1}_{t}(M)\supseteq\varphi^{2}_{t}(M)$, we take
$\overline{a}\in{}^{\lg(\overline{x})}M$ be such that
$M\models\varphi^{2}_{t}(\overline{a})$ and shall show that
$M\models\varphi^{1}_{t}(\overline{a})$. Similar to the Case (3), we can write

$\varphi^{\ell}_{t}(\overline{x}):=(\forall^{\sigma}\overline{x}^{\prime})\varphi^{\ell}_{s}(\overline{x}_{s},\overline{x}^{\prime})\quad\ell=1,2.$

Suppose it is not the case that $M\models\varphi^{1}_{t}(\overline{a})$. This
means, by Definition 2.10(2)(b), that there are pairwise disjoint
$\overline{b}_{\zeta}\in{}^{\lg(\overline{x}^{\prime})}M$, for $\zeta<\sigma$
such that

$M\models\neg\varphi^{1}_{s}(\overline{a},\overline{b}_{\zeta}).$

By the induction hypothesis, we have
$\varphi^{1}_{s}(M)\supseteq\varphi^{2}_{s}(M)$, hence for each
$\zeta<\sigma$,

$M\models\neg\varphi^{2}_{s}(\overline{a},\overline{b}_{\zeta}).$

The later means that $M\models\varphi^{2}_{t}(\overline{a})$ is not true,
which contradicts our initial assumption. This completes the proof of clause
(4)(a).

Finally, let us turn to check the property presented in clause (4)(b) from
Definition 3.1. Suppose $\varphi_{\alpha,\beta}(\overline{x})\in\mathscr{L}$
for $\alpha<\beta<\lambda$ are given. We need to find some
$\alpha_{1}<\alpha_{2}<\alpha_{3}<\kappa$ such that

$\varphi_{\alpha_{1},\alpha_{2}}\geq\varphi_{\alpha_{1},\alpha_{3}},\varphi_{\alpha_{2},\alpha_{3}}.$

To see this, we define a coloring $\mathbf{c}:[\lambda]^{3}\to 2\times 2$ as
follows. Fix $\alpha<\beta<\gamma<\lambda$, and define the following pairing
function

$\mathbf{c}(\\{\alpha,\beta,\gamma\\}):=\bigg{(}\emph{truth value of
}\varphi_{\alpha,\beta}\geq\varphi_{\alpha,\gamma},\emph{ truth value of
}\varphi_{\alpha,\gamma}\geq\varphi_{\beta,\gamma}\bigg{)}.$

By the assumption $\lambda\to(\kappa+1)_{4}^{3}$, there is $X\subseteq\lambda$
of order type $\kappa+1$ such that $\mathbf{c}\restriction[X]^{3}$ is
constant. Let $\alpha_{i}\in X$, for $i\leq\kappa$, be an increasing
enumeration of $X$. Consider the sequence

$\\{\varphi_{\alpha_{0},\alpha_{i}}:i<\kappa\\},$

and applying clause (4)(a) to it. This gives us $i<j<\kappa$ such that

$\varphi_{\alpha_{0},\alpha_{i}}\geq\varphi_{\alpha_{0},\alpha_{j}}.$

Note that this implies that
$\mathbf{c}(\alpha_{0},\alpha_{i},\alpha_{j})=(1,\iota)$, for some
$\iota\in\\{0,1\\}.$ Since $\mathbf{c}\restriction[X]^{3}$ is constant, it
follows that for any $\alpha<\beta<\gamma$ from $X$, we have
$\mathbf{c}(\alpha,\beta,\gamma)=(1,\iota)$, in particular

  * $(*)_{1}$

$\varphi_{\alpha,\beta}\geq\varphi_{\alpha,\gamma}$, for all
$\alpha<\beta<\gamma$ in $X$.

Again, applying clause (4)(a) to the sequence
$\\{\varphi_{\alpha_{i},\alpha_{\kappa}}:i<\kappa\\},$ we can find some
$i<j<\kappa$ such that

$\varphi_{\alpha_{i},\alpha_{\kappa}}\geq\varphi_{\alpha_{j},\alpha_{\kappa}}.$

It follows that $\mathbf{c}(\alpha_{i},\alpha_{j},\alpha_{\kappa})=(1,1)$,
hence as $\mathbf{c}\restriction[X]^{3}$ is constant, we have
$\mathbf{c}(\alpha,\beta,\gamma)=(1,1)$, for all $\alpha<\beta<\gamma$ from
$X$. In particular,

  * $(*)_{2}$

$\varphi_{\alpha,\gamma}\geq\varphi_{\beta,\gamma}$, for any
$\alpha<\beta<\gamma$ in $X$.

Now, combining $(*)_{1}$ along with $(*)_{2}$, we get that for all
$\alpha<\beta<\gamma$ from $X$

$\varphi_{\alpha,\beta}\geq\varphi_{\alpha,\gamma}\geq\varphi_{\beta,\gamma},$

and this completes the proof of (i).

(ii): This is similar to case (i). ∎

######  Remark 3.15.

By the Erdos-Rado partition theorem, see [11], it suffices to take
$\lambda=\beth_{2}(\kappa)^{+}$.

##  4\. $\kappa$-algebraic closure

In this section, and among other things, we define the concept of closure of a
sequence inside an additive model and present some properties of it.

######  Definition 4.1.

Suppose $\mathbf{f}=(M,\mathscr{L},\lambda,\kappa,\theta,\Omega)$ is an
additive frame, $\varepsilon<\theta$ and $\overline{a}\in{}^{\varepsilon}M$.
We define the closure of $\overline{a}$ in $M$ as the following:

$\rm{cl}(\overline{a},M):=\\{b\in
M:\varphi_{{\overline{a}^{\smallfrown}\langle b\rangle}}(M,\overline{a})\
\text{has cardinality}\ <\kappa\\}.$

In what follows we will use the following result several times:

######  Lemma 4.2.

Suppose $\mathbf{f}=(M,\mathscr{L},\lambda,\kappa,\theta,\Omega)$ is a general
frame as in Theorem 3.14, $\varepsilon<\theta$ and
$\overline{a}\in{}^{\varepsilon}M$. We assume in addition that
$\kappa\in\Omega$ and $\mathscr{L}$ is closed under $\exists^{\kappa}$. Then
the following three assertions are true:

  1. (A)

the set $\rm{cl}(\overline{a},M)$ has cardinality $<\kappa.$

  2. (B)

$\\{a_{i}:i<\varepsilon\\}\subseteq\rm{cl}(\overline{a},M).$

  3. (C)

Assume $b\in\rm{cl}(\overline{a},M)$. Then
$\rm{cl}(\overline{a}^{\frown}\langle
b\rangle,M)\subseteq\rm{cl}(\overline{a},M)$.

###### Proof.

(a). Suppose not, and let $\overline{a}\in{}^{\varepsilon}M$ be such that the
set $\rm{cl}(\overline{a},M)$ has cardinality $\geq\kappa.$ This gives us a
family $\\{b_{\alpha}:\alpha<\kappa\\}\subseteq\rm{cl}(\overline{a},M)$.
Define $\overline{a}_{\alpha}:=\overline{a}^{\smallfrown}\langle
b_{\alpha}\rangle$. In the light of Lemma 3.8, there is $\alpha_{*}<\kappa$
such that the set

$X:=\\{\beta<\kappa:\overline{a}_{\beta}\in\varphi_{\overline{a}_{\alpha_{*}}}(M)\\}$

is unbounded in $\kappa.$ So, $\\{\overline{a}_{\beta}:\beta\in
X\\}\subseteq\varphi_{\overline{a}_{\alpha_{*}}}(M,\overline{a})$, which
implies that

$|\varphi_{\overline{a}_{\alpha_{*}}}(M,\overline{a})|\geq|X|=\kappa.$

This contradicts the fact that $b_{\alpha_{*}}\in\rm{cl}(\overline{a},M).$

(b). Let $i<\varepsilon$, and define the formula $\psi(\overline{x},y)$ as

$\psi(\overline{x},y):=(y=x_{i}).$

Then $\psi(\overline{a},M)=\\{a_{i}\\}$. It is also clear that
$\psi(\overline{x},y)$ is minimal with this property. This implies that

$\varphi_{\overline{a}^{\frown}\langle
a_{i}\rangle}(\overline{a},M)=\psi(\overline{a},M).$

In particular, $|\varphi_{\overline{a}^{\frown}\langle
a_{i}\rangle}(\overline{a},M)|=1<\kappa$, and consequently
$a_{i}\in\rm{cl}(\overline{a},M).$

(c). Suppose $d\in\rm{cl}(\overline{a}^{\frown}\langle b\rangle,M)$. Thanks to
Definition 4.1, $\varphi_{\overline{a}^{\frown}\langle
b\rangle^{\smallfrown}\langle d\rangle}(\overline{a},b,M)$ has cardinality
less that $\kappa.$ As $b\in\rm{cl}(\overline{a},M)$, clearly

$(\ast)_{1}\quad\quad\quad\emph{ the set
}B:=\varphi_{{\overline{a}^{\smallfrown}\langle
b\rangle}}(\overline{a},M)\emph{ has cardinality }<\kappa.$

For $b_{1}\in B$ let $A_{b_{1}}:=\varphi_{\overline{a}^{\frown}\langle
b\rangle^{\smallfrown}\langle d\rangle}(\overline{a},b_{1},M)$.

We now show that

$(\ast)_{2}\qquad\quad\quad\quad\emph{ if }b_{1}\in B\emph{ then
}A_{b_{1}}\emph{ has cardinality }<\kappa.$

Assume towards a contradiction that $|A_{b_{1}}|\geq\kappa$ for some $b_{1}\in
B$. Reformulating this, means that:

$M\models\varphi_{\overline{a}^{\frown}\langle
b\rangle}(\overline{a},b_{1})\wedge\exists^{\kappa}z\
\varphi_{\overline{a}^{\frown}\langle b\rangle^{\smallfrown}\langle
d\rangle}(\overline{a},b_{1},z).$

Let

$\psi(\overline{x},{y}):=\varphi_{\overline{a}^{\frown}\langle
b\rangle}(\overline{x},{y})\wedge\exists^{\kappa}z\
\varphi_{\overline{a}^{\frown}\langle b\rangle^{\smallfrown}\langle
d\rangle}(\overline{x},{y},z),$

and recall that $\psi\in\mathscr{L}$ and $\overline{a}^{\frown}\langle
b_{1}\rangle\in\psi(M)$. Note that $\overline{a}^{\frown}\langle
b\rangle\notin\psi(M)$, as

$|\varphi_{\overline{a}^{\frown}\langle b\rangle^{\smallfrown}\langle
d\rangle}(\overline{a},b,M)|<\kappa.$

Next we bring the following claim:

$(\ast)_{2.1}\quad\quad\quad\overline{a}^{\frown}\langle
b_{1}\rangle\notin\varphi_{\overline{a}^{\frown}\langle b\rangle}(M).$

To see this we argue by the way of contradiction that
$\overline{a}^{\frown}\langle
b_{1}\rangle\in\varphi_{\overline{a}^{\frown}\langle b\rangle}(M)$. This
implies following the minimality condition that

$\varphi_{\overline{a}^{\frown}\langle
b\rangle}(M)\subseteq\varphi_{\overline{a}^{\frown}\langle
b_{1}\rangle}(M)\subseteq\psi(M).$

Consequently, $\overline{a}^{\frown}\langle b\rangle\in\psi(M)$. This
contradiction completes the proof of $(\ast)_{2.1}.$ But, we have $\langle
b_{1}\rangle\in\varphi_{\overline{a}^{\frown}\langle
b\rangle}(\overline{a},M)$. This yields that

$(\ast)_{2.2}\quad\quad\quad\overline{a}^{\frown}\langle
b_{1}\rangle\in\varphi_{\overline{a}^{\frown}\langle b\rangle}(M).$

But $(\ast)_{2.1}$ and $(\ast)_{2.2}$ together lead to a contradiction. In
sum, the desired property $(\ast)_{2}$ is valid.

Recalling that $\kappa$ is regular, it follows that

$(\ast)_{3}\qquad\quad\bigcup_{b_{1}\in\beta}\varphi_{\overline{a}^{\frown}\langle
b\rangle^{\smallfrown}\langle d\rangle}(\overline{a},b_{1},M)\emph{ has
cardinality }<\kappa.$

We will show that

$(\dagger)\quad\quad\quad\quad\quad\varphi_{{\overline{a}^{\smallfrown}\langle
d\rangle}}(\overline{a},M)\subseteq\bigcup_{b_{1}\in
B}\varphi_{\overline{a}^{\frown}\langle b\rangle^{\smallfrown}\langle
d\rangle}(\overline{a},b_{1},M),$

from which it will follow that $\varphi_{{\overline{a}^{\smallfrown}\langle
d\rangle}}(\overline{a},M)$ has cardinality less that $\kappa,$ and hence by
definition, $d\in\rm{cl}(\overline{a},M)$. Let us prove $(\dagger)$. To this
end, let $d_{1}\in\varphi_{{\overline{a}^{\smallfrown}\langle
d\rangle}}(\overline{a},M)$. This implies that $\overline{a}^{\frown}\langle
d_{1}\rangle\in\varphi_{{\overline{a}^{\smallfrown}\langle d\rangle}}(M)$.
Clearly,

$M\models\exists y\varphi_{\overline{a}^{\frown}\langle
b\rangle^{\smallfrown}\langle d\rangle}(\overline{a},y,d),$

hence,

$M\models\exists y\varphi_{\overline{a}^{\frown}\langle
b\rangle^{\smallfrown}\langle d\rangle}(\overline{a},y,d_{1}).$

This gives us some $b_{1}$ so that

$M\models\varphi_{\overline{a}^{\frown}\langle b\rangle^{\smallfrown}\langle
d\rangle}(\overline{a},b_{1},d_{1}).$

Then ${b_{1}\in B}$, and $d_{1}\in\varphi_{\overline{a}^{\frown}\langle
b\rangle^{\smallfrown}\langle d\rangle}(\overline{a},b_{1},M),$ and
consequently, $(\dagger)$ holds. We are done. ∎

######  Definition 4.3.

Suppose $\mathbf{f}=(M,\mathscr{L},\lambda,\kappa,\theta,\Omega)$ is a general
frame. For $\overline{a}\in{}^{<\theta}M$, we introduce the following:

  1. (A)

$\rm{afn}(b,\overline{a})=\\{c\in\rm{cl}(\overline{a},M):\varphi_{\overline{a}^{\smallfrown}\langle
b\rangle}\leq\varphi_{\overline{a}^{\smallfrown}\langle c\rangle}\\}.$

  2. (B)

Suppose $\mathbf{f}$ is abelian. Then
$\rm{grp}(b,\overline{a})=\\{c_{1}-c_{2}:c_{1},c_{2}\in\rm{afn}(b,\overline{a})\\}.$

######  Hypothesis 4.4.

In what follows, and up to the end of this section, let us assume that
$\mathbf{f}=(M,\mathscr{L},\lambda,\kappa,\theta,\Omega)$ is an additive
frame, $\kappa\in\Omega$ and $\mathscr{L}$ is closed under
$\exists^{\kappa}x$.

######  Lemma 4.5.

Let $\overline{a}\in{}^{<\theta}M$ and $b\in\rm{cl}(\overline{a},M)$. Then
$\rm{afn}(b,\overline{a})$ is a subset of $\rm{cl}(\overline{a})$ and it is
affine.

###### Proof.

First, recall that
“$\rm{afn}(b,\overline{a})\subseteq\rm{cl}(\overline{a},M)$” holds by the
definition. For the second phrase, we have to show that
$\rm{afn}(b,\overline{a})$ is closed under $x-y+z$. To see this, let
$c_{i}\in\rm{afn}(b,\overline{a})$ for $i=1,2,3$ and set
$c:=c_{1}-c_{2}+c_{3}.$ Since $\varphi_{\overline{a}^{\smallfrown}\langle
b\rangle}(M)$ is affine-closed,

$\overline{a}^{\smallfrown}\langle c\rangle=\overline{a}^{\smallfrown}\langle
c_{1}\rangle-\overline{a}^{\smallfrown}\langle
c_{2}\rangle+\overline{a}^{\smallfrown}\langle
c_{3}\rangle\in\varphi_{\overline{a}^{\smallfrown}\langle b\rangle}(M).$

According to the minimality, $\varphi_{\overline{a}^{\smallfrown}\langle
b\rangle}\leq\varphi_{\overline{a}^{\smallfrown}\langle c\rangle}$. Thanks to
Lemma 4.2, $c\in\rm{cl}(\overline{a},M)$. Hence
$c\in\rm{afn}(b,\overline{a})$, and we are done. ∎

######  Lemma 4.6.

The following holds:

  1. (A)

$\rm{grp}(b,\overline{a})$ is a subgroup of $M.$

  2. (B)

$\rm{afn}(b,\overline{a})=\\{b+d:d\in\rm{grp}(b,\overline{a})\\}.$

###### Proof.

For clause (1), let $c_{i}\in\rm{grp}(b,\overline{a}),$ where $i=1,2$.
Following definition, there are some
$b_{i,1},b_{i,2}\in\rm{afn}(b,\overline{a})$ such that
$c_{i}=b_{i,1}-b_{i,2}$. So,

$\begin{array}[]{ll}c_{1}-c_{2}&=(b_{1,1}-b_{1,2})-(b_{2,1}-b_{2,2})\\\
&=(b_{1,1}-b_{1,2}+b_{2,2})-b_{2,1}.\end{array}$

According to Lemma 4.5, we know
$b^{*}_{2,1}=b_{1,1}-b_{1,2}+b_{2,2}\in\rm{afn}(b,\overline{a})$, and hence by
definition of $\rm{grp}(b,\overline{a})$,

$c_{1}-c_{2}=b^{*}_{2,1}-b_{2,1}\in\rm{grp}(b,\overline{a}).$

To prove clause (2), let $c\in\rm{afn}(b,\overline{a})$. As clearly
$b\in\rm{afn}(b,\overline{a}),$ by clause (1),
$-b+c\in\rm{grp}(b,\overline{a}),$ hence

$c=b+(-b+c)\in\big{\\{}b+d:d\in\rm{grp}(b,\overline{a})\big{\\}}.$

Conversely, suppose $d\in\rm{grp}(b,\overline{a})$. Due to its definition,
there are for some $c_{1},c_{2}\in\rm{afn}(b,\overline{a})$ so that
$d=c_{1}-c_{2}$. Consequently, in view of Lemma 4.5, we see

$b+d=b-c_{2}+c_{1}\in\rm{afn}(b,\overline{a}).$

The equality follows. ∎

######  Definition 4.7.

Let $\mathbf{f}=(M,\mathscr{L},\lambda,\kappa,\theta,\Omega)$ be an additive
frame such that $\kappa\in\Omega$ and $\mathscr{L}$ is closed under
$\exists^{\kappa}x$. We say $\mathbf{f}$ is very nice, if it satisfies the
following extra properties:

  1. (A)

$\mathscr{L}_{\mathbf{f}}=\mathscr{L}_{\infty,\theta}^{\rm{pe}}(\tau_{M})$, or
just $\mathscr{L}_{\mathbf{f}}$ is closed under $\exists\overline{x}_{u},$ up
to equivalence, where $|u|<\theta$.

  2. (B)

For every $X\subseteq{}^{\varepsilon}M$, there is a formula
$\varphi_{X}(\overline{x})$, such that $X\subseteq\varphi_{X}(M)$, and if
$\psi(\overline{x})$ is such that $X\subseteq\psi(M),$ then
$\varphi_{X}(M)\subseteq\psi(M)$.

######  Discussion 4.8.

By a repetition of the argument presented in the proof of Theorem 3.14 we know
Definition 4.7(2) holds, when $M$ is an $R$-module and $\mathbf{f}$ is defined
as in 3.14.

######  Proposition 4.9.

Suppose the additive frame
$\mathbf{f}=(M,\mathscr{L},\lambda,\kappa,\theta,\Omega)$ is very nice. The
following conditions hold.

  1. (A)

Assume $b\in M$ and $\overline{a}\in{}^{\varepsilon}M$. There exists some
formula $\varphi(\overline{x},y)$ with
$\operatorname{lg}(\overline{x})=\varepsilon$ such that
$\varphi(\overline{a},M)=\rm{grp}(b,\overline{a}).$

  2. (B)

If $c\in\rm{grp}(b,\overline{a})$ and $b\in\rm{cl}(\overline{a})$ then
$c\in\rm{cl}(\overline{a}).$ Moreover, $\rm{cl}(\overline{a})$ is a subgroup
of $M$.

  3. (C)

Let $b\in\rm{cl}(\overline{a})$. Then $\varphi_{\rm{grp}(b,\overline{a})}(M)$
is of cardinality $<\kappa.$

###### Proof.

(1): Define the formula $\varphi$ as

$\varphi(\overline{x},y)=(\exists{y}^{1},y^{2})\bigg{[}\varphi_{\overline{a}^{\smallfrown}\langle
b\rangle}(\overline{x},y^{1})\wedge\varphi_{\overline{a}^{\smallfrown}\langle
b\rangle}(\overline{x},y^{2})\wedge y=y_{2}-y_{1}\bigg{]}.$

We show that $\varphi$ is as required. By Definition 4.7,
$\varphi(\overline{x},y)\in\mathscr{L}.$

First, suppose that $b\in\rm{grp}(b,\overline{a})$, and let
$b_{1},b_{2}\in\rm{afn}(b,\overline{a})$ be such that $b=b_{2}-b_{1}$. Then
$b_{1},b_{2}$ witness $M\models\varphi(\overline{a},b).$ Hence

$(\ast)_{1}\quad\quad\qquad\rm{grp}(b,\overline{a})\subseteq\varphi(\overline{a},M).$

In order to prove the reverse inclusion, suppose that
$b\in\varphi(\overline{a},M).$ This implies that
$M\models\varphi(\overline{a},b)$. Take $b_{1},b_{2}$ be witness it, i.e.,
$b=b_{2}-b_{1}$ and

$M\models\varphi_{\overline{a}^{\smallfrown}\langle
b\rangle}(\overline{a},b_{1})\wedge\varphi_{\overline{a}^{\smallfrown}\langle
b\rangle}(\overline{a},b_{2}).$

On the other hand, for $l=1,2$ we have

$M\models\varphi_{\overline{a}^{\smallfrown}\langle
b\rangle}(\overline{a},b_{l})\Rightarrow\varphi_{\overline{a}^{\smallfrown}\langle
b\rangle}\leq\varphi_{\overline{a}^{\smallfrown}\langle b_{l}\rangle},$

hence $b_{l}\in\rm{afn}(b,\overline{a})$. Consequently,
$b=b_{2}-b_{1}\in\rm{grp}(b,\overline{a}).$ As $b$ was arbitrary, we conclude
that

$(\ast)_{2}\quad\qquad\quad\varphi(\overline{a},M)\subseteq\rm{afn}(b,\overline{a}).$

By $(\ast)_{1}$ and $(\ast)_{2}$, we have
$\varphi(M,\overline{a})=\rm{afn}(b,\overline{a})$, and we are done.

(2): In the light of Lemma 4.5, it suffices to show that
$\rm{cl}(\overline{a})$ is a subgroup of $M$. To this end, let
$b_{1},b_{2}\in\rm{cl}(\overline{a}),\,\varepsilon=\rm{lg}(\overline{a})$ and
let $\varphi(\overline{x},y)$ be as in clause (1). It is easily seen that:

  1. $\bullet_{1}$

$M\models\varphi(\overline{a},b_{2}-b_{1}),$

  2. $\bullet_{2}$

$\varphi(\overline{a},M)=\\{b^{\prime\prime}-b^{\prime}:b^{\prime\prime}\in\varphi_{\overline{a}^{\smallfrown}\langle
b_{2}\rangle}(\overline{a},M)\ \text{and}\
b^{\prime}\in\varphi_{\overline{a}^{\smallfrown}\langle
b_{1}\rangle}(\overline{a},M)\\},$

  3. $\bullet_{3}$

$\varphi(\overline{a},M)$ has cardinality $<\kappa.$

Thanks to the minimality condition,

$\varphi_{\overline{a}^{\smallfrown}\langle
b_{2}-b_{1}\rangle}(M)\subseteq\varphi(\overline{a},M).$

In other words, $|\varphi_{\overline{a}^{\smallfrown}\langle
b_{2}-b_{1}\rangle}(M)|<\kappa$, which implies that
$b_{2}-b_{1}\in\rm{cl}(\overline{a}).$ We have proved

$b_{1},b_{2}\in\rm{cl}(\overline{a})\Rightarrow
b_{2}-b_{1}\in\rm{cl}(\overline{a}).$

Therefore, $\rm{cl}(\rm{\overline{a}})$ is a subgroup of $M.$

(3): The proof is similar to the proof of clause (2). ∎

##  5\. Dichotomy behavior of absolutely co-Hopfian property

This section is devoted to the proof of Theorem 1.3 from the introduction. We
start by recalling the definition of (co)-Hopfian modules.

######  Definition 5.1.

Let $M$ be an $R$-module.

  * (i)

$M$ is called _Hopfian_ if its surjective $R$-endomorphisms are automorphisms.

  * (ii)

$M$ is called _co-Hopfian_ if its injective $R$-endomorphisms are
automorphisms.

This can be extended to:

######  Definition 5.2.

Let $M$ be a $\tau$-model.

  * (i)

$M$ is called _Hopfian_ if its surjective $\tau$-morphisms are
$\tau$-automorphisms.

  * (ii)

$M$ is called _co-Hopfian_ if its injective $\tau$-morphisms are
$\tau$-automorphisms.

For the convenience of the reader, we present the definition of potentially
isomorphic, and discuss some basic facts about them which are used in the
paper, and only sketch the proofs in most instances.

######  Definition 5.3.

Let $M,N$ be two structures of our vocabulary. Recall that $M$ and $N$ are
called potentially isomorphic provided they are isomorphic in some forcing
extension.

Recall that a group $G$ is called absolutely co-Hopfian (resp. Hopfian) if it
is co-Hopfian (resp. Hopfian) in any further generic extension of the
universe.

######  Discussion 5.4.

Suppose $M$ and $N$ are potentially isomorphic. According to [19] and [20]
this is holds iff for every $\mathscr{L}_{\infty,\aleph_{0}}$-sentence
$\varphi,$

$(M\models\varphi)\Longleftrightarrow(N\models\varphi).$

We denote this property by $M\equiv_{\mathscr{L}_{\infty,\aleph_{0}}}N.$

The following is a simple variant of Discussion 5.4. We state it in our
context.

######  Lemma 5.5.

Assume $M$ and $N$ are two $\tau$-structures.

  1. (A)

Suppose for every sentence $\varphi\in\mathscr{L}_{\infty,\aleph_{0}}(\tau)$,

$(M\models\varphi)\Longrightarrow(N\models\varphi).$

Then there is an embedding of $M$ into $N$ in $V[G_{\mathbb{P}}]$, where
$\mathbb{P}$ collapses $|M|+|N|$ into $\aleph_{0}$.

  2. (B)

In clause (1), it suffices to consider sentences $\varphi$ in the closure of
base formulas under arbitrary conjunctions and $\exists x.$

###### Proof.

We give a proof for completeness. Let $M=\\{a_{n}:n<\omega\\}$ be an
enumeration of $M$ in $V[G_{\mathbb{P}}]$. By induction on $n$ we define a
sequence $\langle b_{n}:n<\omega\rangle$ of members of $N$ such that for each
formula $\varphi(x_{0},\cdots,x_{n})$ from $\mathscr{L}_{\infty,\aleph_{0}}$,

$(*)_{n}\quad\quad\quad\big{(}M\models\varphi(a_{0},\cdots,a_{n})\big{)}\Rightarrow\big{(}N\models\varphi(b_{0},\cdots,b_{n})\big{)}.$

Let

$\Phi_{0}(x_{0})=\bigwedge\\{\varphi(x_{0}):M\models\varphi(a_{0})\\}\in\mathscr{L}_{\infty,\aleph_{0}}.$

Then $M\models\Phi_{0}(a_{0})$, and by our assumption, there exists some
$b_{0}\in N$ such that $N\models\Phi_{0}(b_{0})$. Now suppose that $n<\omega$
and we have defined $b_{0},\cdots,b_{n}.$ We are going to define $b_{n+1}$.
Let

$\Phi_{n+1}(x_{0},\ldots,x_{n+1})=\bigwedge\big{\\{}\varphi(x_{0},\ldots,x_{n+1}):M\models\varphi(a_{0},\cdots,a_{n+1})\big{\\}}.$

Clearly, $\Phi_{n+1}(x_{0},\cdots,x_{n+1})\in\mathscr{L}_{\infty,\aleph_{0}}$.
Also,

$M\models\exists x_{n+1}\Phi_{n+1}(a_{0},\cdots,a_{n},x_{n+1}).$

According to the induction hypothesis $(*)_{n}$, we have

$N\models\varphi(b_{0},\cdots,b_{n+1})$

for some $b_{n+1}\in N$. This completes the construction of the sequence
$\langle b_{n}:n<\omega\rangle$. The assignment $a_{n}\mapsto b_{n}$ defines a
map $f:M\to N$ which is an embedding of $M$ into $N$. ∎

######  Fact 5.6.

  1. (A)

Let $\lambda\geq\kappa_{\rm{beau}}$ and let $\langle
G_{\alpha}:\alpha<\lambda\rangle$ be a sequence of $\tau$-models with
$|\tau|<\kappa_{\rm{beau}}(\tau),$. Then in some forcing extension
$\mathbf{V}^{\mathbb{P}},G_{\alpha}$ is embeddable into $G_{\beta},$ for some
$\alpha<\beta<\lambda.$ Here, $\mathbb{P}$ collapses
$|G_{\alpha}|+|G_{\beta}|$ into $\aleph_{0}$. Moreover, if $x_{\gamma}\in
G_{\gamma}$ for $\gamma<\lambda$ then for some $\alpha<\beta<\lambda,$ in some
$\mathbf{V}^{\mathbb{P}}$ there is an embedding of $G_{\alpha}$ into
$G_{\beta}$ mapping $x_{\alpha}$ to $x_{\beta}$222This explains why [14] gets
only indecomposable abelian groups (not endo-rigid)..

  2. (B)

Suppose $M$ and $N$ are abelian groups, or $R$-modules, and for every sentence
$\varphi\in\mathscr{L}_{\infty,\theta}^{\rm{ce}}(\tau)$, we have

$(M\models\varphi)\Longrightarrow(N\models\varphi).$

Then there is an embedding of $M$ into $N$ in $V[G_{\mathbb{P}}]$, where
$\mathbb{P}$ collapses $|M|+|N|$ into $\aleph_{0}$.

  3. (C)

Moreover, we can strengthen the conclusion of part (2) to the following:

     * $(\ast)$

there is a $\mathbb{P}$-name $\pi$ satisfying:

       * $(\ast)_{1}$

If $\overline{a}\in({}^{<\theta}M)\cap V$ then
$\pi(\overline{a})\in({}^{<\theta}N)\cap V$,

       * $(\ast)_{2}$

$\Vdash_{\mathbb{P}}\pi$ maps $\overline{a}\in({}^{<\theta}M)\cap V$ onto
$\\{\overline{b}\in{}^{<\theta}M:\mathrm{rang}(\overline{b})\subseteq\mathrm{rang}(\pi)\\}.$

###### Proof.

For (1), see [12]. Parts (2) and (3) are standard, see for example [19]. ∎

Now, we are ready to prove:

######  Theorem 5.7.

The following assertions are valid:

  1. (A)

If $M$ is an abelian group of cardinality $\geq\kappa:=\kappa_{\rm{beau}}$,
then $M$ is not absolutely co-Hopfian, indeed, after collapsing the size of
$M$ into $\omega$, there is a one-to-one endomorphism $\varphi\in\rm{End}(M)$
which is not onto.

  2. (B)

If $M$ is an $R$-module of cardinality $\geq\kappa=\kappa_{\rm{beau}}(R)$,
then $M$ is not absolutely co-Hopfian.

  3. (C)

If $M$ is an $\tau$-model of cardinality
$\geq\kappa=\kappa_{\rm{beau}}(\tau)$, then $M$ is not absolutely co-Hopfian.

###### Proof.

Let $M$ be an abelian group or an $R$-module of size $|M|\geq\kappa$. Thanks
to Theorem 3.14, there exists an additive frame $\mathbf{f}$ as there such
that $M:=M_{\mathbf{f}}$,
$\mathscr{L}_{\mathbf{f}}=\mathscr{L}_{\infty,\theta}^{\rm{ce}}(\tau),\kappa_{\mathbf{f}}=\kappa$,
$\lambda_{\mathbf{f}}=\beth_{2}(\kappa)^{+}$ and
$\theta_{\mathbf{f}}=\aleph_{0}.$

The proof splits into two cases:

Case 1: for some $\varepsilon<\theta,$ and
$\overline{a},\overline{b}\in{}^{\varepsilon}M$,
$\varphi_{\overline{a}}(M)\supsetneqq\varphi_{\overline{b}}(M)$.

Consider the $\tau$-models $(M,\overline{b})$ and $(M,\overline{a})$. Let
$\varphi\in\mathscr{L}_{\infty,\theta}^{\rm{ce}}(\tau)$ be a sentence, and
suppose that $(M,\overline{a})\models\varphi.$ As
$\varphi_{\overline{a}}(M)\supsetneqq\varphi_{\overline{b}}(M)$, it follows
that $(M,\overline{b})\models\varphi.$ Thus by Fact 5.6(2), working in the
generic extension by $\mathbb{P}=\text{Col}(\aleph_{0},|M|)$, there exists a
one-to-one endomorphism $\pi\in\rm{End}(M)$ such that
$\pi(\overline{a})=\overline{b}.$

There is nothing to prove if $\pi$ is not onto. So, without loss of generality
we may and do assume that $\pi$ is onto. Then $\pi,\pi^{-1}\in\rm{Aut}(M)$ and
$\pi^{-1}(\overline{b})=\overline{a}$. We claim that
$\varphi_{\overline{a}}\leq\varphi_{\overline{b}}$. Due to the minimality
condition for $\varphi_{\overline{a}}$, it is enough to show that
$\overline{a}\in\varphi_{\overline{b}}(M)$. To this end, recall that
$\overline{b}\in\varphi_{\overline{b}}(M)$. By definition,
$M\models\varphi_{\overline{b}}[\overline{b}]$. In the light of Lemma 2.15(2)
we observe that

$M\models\varphi_{\overline{b}}[\pi^{-1}(\overline{b})]=\varphi_{\overline{b}}[\overline{a}].$

By definition, this means that $\overline{a}\in\varphi_{\overline{b}}(M)$, as
requested. Consequently,
$\varphi_{\overline{a}}(M)\subseteq\varphi_{\overline{b}}(M)$, which
contradicts our assumption.

Case 2: not case $1$.

Given two sets $A,B\subseteq M$, by $A\|B$ we mean that $A\subseteq B$ or
$B\subseteq A$. So in this case, the following holds:

$(*)$: $\forall\varepsilon<\kappa$ and
$\forall\overline{a},\overline{b}\in{}^{\varepsilon}M$ we have
$\bigg{(}\varphi_{\overline{a}}(M)\|\varphi_{\overline{b}}(M)\Rightarrow\varphi_{\overline{a}}(M)=\varphi_{\overline{b}}(M)\bigg{)}.$

Set $\Gamma=\\{\varphi_{\overline{a}}:\overline{a}\in{}^{<\theta}M\\}$. Now,
we have the following easy claim.

######  Claim 5.8.

$\Gamma$ is a set of cardinality $<\kappa.$

###### Proof.

To see this, set
$\Gamma_{\varepsilon}:=\\{\varphi_{\overline{a}}:\overline{a}\in{}^{\varepsilon}M\\}$.
Clearly, $\Gamma=\bigcup_{\varepsilon<\theta}\Gamma_{\varepsilon}$, and since
$\theta<\kappa$ and $\kappa$ is regular, it suffices to show that
$|\Gamma_{\varepsilon}|<\kappa$ for all $\varepsilon<\theta$. Suppose not and
search for a contradiction. Take $\varepsilon<\theta$ be such that
$|\Gamma_{\varepsilon}|\geq\kappa$. This enables us to find a sequence
$\langle\overline{a}_{\alpha}:\alpha<\kappa\rangle$ in ${}^{\varepsilon}M$
such that

  * $\bullet_{1}$

$\forall{\alpha}<\kappa,~{}\varphi_{{\overline{a}}_{\alpha}}\in\Gamma_{\varepsilon}$

  * $\bullet_{2}$

$\forall\alpha\neq\beta$,
$\varphi_{{\overline{a}}_{\alpha}}(M)\neq\varphi_{{\overline{a}}_{\beta}}(M)$.

We apply the property presented in Definition 3.1(5)(a) to the family
$\\{\varphi_{{\overline{a}}_{\alpha}}\\}_{\alpha<\kappa}$, to find some
$\alpha<\beta<\kappa$ such that
$\varphi_{{\overline{a}}_{\beta}}(M)\supseteq\varphi_{{\overline{a}}_{\alpha}}(M).$
By $(\ast)$, this implies that
$\varphi_{{\overline{a}}_{\beta}}(M)=\varphi_{{\overline{a}}_{\alpha}}(M),$
which contradicts $\bullet_{2}$. ∎

Let $\chi:=|M|\geq\kappa$, and let $\mathbb{P}:=\rm{Col}(\aleph_{0},\chi)$.
Forcing with $\mathbb{P}$, collapses $|M|$ into $\aleph_{0}$, i.e., for any
$\mathbb{P}$-generic filter $G_{\mathbb{P}}$ over $V$, we have

$V[G_{\mathbb{P}}]\models$“$M$ is countable”.

We are going to show there is in $V[G_{\mathbb{P}}]$, there exists a 1-1 map
$\pi:M\to M$ which is not surjective. To this end, we define approximations to
the existence of such $\pi$:

  1. $\boxplus$

Let $\mathrm{AP}$ be the set of all triples $(\overline{a},\overline{b},c)$
such that:

    1. (a)

$\overline{a},\overline{b}\in{}^{\varepsilon}M$ for some $\varepsilon<\theta,$

    2. (b)

$\varphi_{\overline{a}}(\overline{x})\equiv\varphi_{\overline{b}}(\overline{x})$
(in $M$),

    3. (c)

$c\in M$ is such that $c\notin\rm{cl}(\overline{b},M),$ i.e.,
$\varphi_{\overline{b}^{\smallfrown}\langle c\rangle}(M)\ \text{has
cardinality}\ \geq\kappa$.

######  Claim 5.9.

$\mathrm{AP}\neq\emptyset.$

###### Proof.

According to Lemma 4.2(a), $\rm{cl}(\overline{a})$ has cardinality $<\kappa.$
In particular, $|\rm{cl}(\emptyset)|<\kappa$, and hence as $|M|\geq\kappa,$ we
can find some $c\in M\setminus\rm{cl}(\emptyset)$, and consequently,
$(\langle\rangle\langle\rangle,c)\in\mathrm{AP}.$ The claim follows. ∎

Next, we bring the following claim, which plays the key role in our proof.

######  Claim 5.10.

Suppose $(\overline{a},\overline{b},c)\in\mathrm{AP}$ and $d_{1}\in M$. Then
there is some $d_{2}\in M$ such that

$\big{(}\overline{a}^{\smallfrown}\langle
d_{1}\rangle,\overline{b}^{\smallfrown}\langle
d_{2}\rangle,c\big{)}\in\mathrm{AP}.$

###### Proof.

Recall that in $M$ we have
$\varphi_{\overline{a}}(\overline{x})\equiv\varphi_{\overline{b}}(\overline{x})$.
First, we use this to find $d\in M$ such that

$(\dagger)\quad\quad\quad M\models\varphi_{\overline{a}^{\smallfrown}\langle
d_{1}\rangle}(\overline{b},d).$

Indeed, we look at the formula

$\psi(\overline{x}):=\exists y\varphi_{\overline{a}^{\smallfrown}\langle
d_{1}\rangle}(\overline{x},y).$

Since $\overline{a}\in\psi(M)$, and due to the minimality of
$\varphi_{\overline{a}}$ with respect to this property, we should have

$\overline{b}\in\varphi_{\overline{b}}(M)=\varphi_{\overline{a}}(M)\subseteq\psi(M).$

In other words,

$M\models\exists y\varphi_{\overline{a}^{\smallfrown}\langle
d_{1}\rangle}(\overline{b},y).$

Hence, for some $d\in M$ we must have
$M\models\varphi_{\overline{a}^{\smallfrown}\langle
d_{1}\rangle}(\overline{b},d).$ So, $(\dagger)$ is proved. From this,
$\overline{b}^{\smallfrown}\langle
d\rangle\in\varphi_{\overline{a}^{\smallfrown}\langle d_{1}\rangle}(M)$. This
implies, using the minimality condition on formulas, that
$\varphi_{\overline{a}^{\smallfrown}\langle
d_{1}\rangle}(M)\subseteq\varphi_{\overline{b}^{\smallfrown}\langle
d\rangle}(M)$. Combining this along with $(*)$ yields that

$\varphi_{\overline{a}^{\smallfrown}\langle
d_{1}\rangle}(M)=\varphi_{\overline{b}^{\smallfrown}\langle d\rangle}(M).$

First, we deal with the case $d\in\rm{cl}(\overline{b},M)$. Thanks to Lemma
4.2(c), and recalling that $\Omega$ contains $\\{1,{\kappa}\\}$, we know
$\rm{cl}(\overline{b}^{\frown}\langle d\rangle)=\rm{cl}(\overline{b})$.
Consequently, $c\notin\rm{cl}(\overline{b}^{\smallfrown}\langle d\rangle)$.
Following definition, one has

$\big{(}\overline{a}^{\smallfrown}\langle
d_{1}\rangle,\overline{b}^{\smallfrown}\langle\rangle,c\big{)}\in\mathrm{AP},$

and we are done by taking $d_{2}=d.$ So, without loss of generality let us
assume that $d\notin\rm{cl}(\overline{b},M).$ It follows that the set

$\mathbf{I}:=\bigg{\\{}c^{\prime}\in
M:M\models\varphi_{\overline{b}^{\smallfrown}\langle
d\rangle}[\overline{b},c^{\prime}]\bigg{\\}}$

has cardinality $\geq\kappa.$ Therefore, there is
$c^{\prime}\in\mathbf{I}\setminus\rm{cl}(\overline{b}^{\smallfrown}\langle
d\rangle,M).$ By the minimality condition on
$\varphi_{\overline{b}^{\smallfrown}\langle d\rangle}$ and since
$\overline{b}^{\smallfrown}\langle
c^{\prime}\rangle\in\varphi_{\overline{b}^{\smallfrown}\langle d\rangle}(M)$,
we have $\varphi_{\overline{b}^{\smallfrown}\langle
d\rangle}(M)\subseteq\varphi_{\overline{b}^{\smallfrown}\langle
c^{\prime}\rangle}(M)$. Now, we use this along with $(*)$ and deduce that

$\varphi_{\overline{b}^{\smallfrown}\langle
d\rangle}(M)=\varphi_{\overline{b}^{\smallfrown}\langle
c^{\prime}\rangle}(M).$

Then, in the same vein as above, we can find some $d^{\prime}$ such that
$\varphi_{\overline{b}^{\smallfrown}\langle
c,d^{\prime}\rangle}(M)=\varphi_{\overline{b}^{\smallfrown}\langle
c^{\prime},d\rangle}(M)$. This yields that
$c\notin\rm{cl}(\overline{b}^{\smallfrown}\langle d^{\prime}\rangle,M)$, hence
$d_{2}=d^{\prime}$ is as required. ∎

In $V[G_{\mathbb{P}}]$, $M$ is countable. Let us list $M$ as
$\\{a_{i}:i<\omega\\}$. We define $\pi:M\to M$ by evaluating $\pi$ at $a_{i}$.
We do this by induction, in such a way that for some fixed $c\in M$ and all
$n<\omega$, if $\pi(a_{i})=b_{i}$, then

$(\dagger\dagger)_{n}\quad\quad\quad\big{(}\langle a_{i}:i<n\rangle,\langle
b_{i}:i<n\rangle,c\big{)}\in\mathrm{AP}.$

Recall from Claim 5.9 and its proof that there is some $c$ in $M$ such that
$(\langle\rangle\langle\rangle,c)\in\mathrm{AP}$. Let us apply Claim 5.10 to

  * $\bullet$

${\overline{a}}:=\langle\rangle$

  * $\bullet$

${\overline{b}}:=\langle\rangle$

  * $\bullet$

$d_{1}:=a_{0}$.

This gives us an element $b_{0}\in M$ such that $\big{(}\langle
a_{0}\rangle,\langle b_{0}\rangle,c\big{)}\in\mathrm{AP}.$ Let
$f(a_{0})=b_{0}$. Now, suppose inductively we have defined $\pi(a_{i})=b_{i}$
for all $i<n$ such that $(\dagger\dagger)_{n}$ is true. Let us apply Claim
5.10 to

  * $\bullet$

${\overline{a}}:=\langle a_{i}:i<n\rangle$,

  * $\bullet$

${\overline{b}}:=\langle b_{i}:i<n\rangle$,

  * $\bullet$

$d_{1}:=a_{n}$.

This gives us an element $b_{n}\in M$ such that

$\big{(}{\overline{a}}^{\smallfrown}\langle
a_{n}\rangle,{\overline{b}}^{\smallfrown}\langle
b_{n}\rangle,c\big{)}\in\mathrm{AP}.$

Let $\pi(a_{n})=b_{n}$, and note that $(\dagger\dagger)_{n+1}$ holds as well.
This completes the inductive definition of $\pi$.

The proof becomes complete if we can show the following three items are
satisfied:

  * $(\maltese)_{1}$

$\pi$ is a homomorphism.

  * $(\maltese)_{2}$

$\pi$ is 1-to-1.

  * $(\maltese)_{3}$

$\pi$ is not surjective.

Let us check these:

  * $(\maltese)_{1}$

Suppose $\varphi=\varphi_{\overline{a}}$ is a first order formula, hence,
without loss of generality, the length of $\overline{a}$ is finite and we can
enlarge $\overline{a}$ to $\langle a_{i}:i<n\rangle$ for some $n<\omega$.
Recall that $b_{i}:=\pi(a_{i})$. By the construction

$\big{(}\langle a_{j}:j<n\rangle,\langle
b_{j}:j<n\rangle,c\big{)}\in\mathrm{AP}.$

Assume $M\models\varphi(\overline{a})$. We show that
$M\models\varphi(\overline{b})$. We have
${\overline{b}}\in\varphi_{\overline{b}}(M)$ and by our construction,
$\varphi_{\overline{b}}(M)=\varphi_{\overline{a}}(M)=\varphi(M)$, hence
${\overline{b}}\in\varphi(M)$, which means that
$M\models\varphi(\overline{b})$. From this, it immediately follows that $\pi$
is a homomorphism.

  * $(\maltese)_{2}$

Following the construction given by Claim 5.10, we can always find $b_{n}$ so
that $b_{n}\neq b_{i}$ for all $i<n$. So, $f$ is 1-to-1.

  * $(\maltese)_{3}$

Suppose by the way of contradiction that $\pi$ is surjective. In particular,
we can find some $n<\omega$ such that $c=\pi(a_{n})=b_{n}$. In the light of
Lemma 4.2(b) we observe that

$c=b_{n}\in\rm{cl}\big{(}\langle b_{i}:i<n+1\rangle,M\big{)},$

which contradicts the following fact

$\big{(}\langle a_{i}:i<n+1\rangle,\langle
b_{i}:i<n+1\rangle,c\big{)}\in\mathrm{AP}.$

The proof is now complete.

(3): The proof of this item is similar to the first part. ∎

##  6\. A new construction of Hopfian groups

In this section we give a new construction of absolutely Hopfian groups. Let
us start with the following well-known definition.

######  Definition 6.1.

A group $G$ is pure in a torsion-free abelian group $H$ if $G\subseteq H$ and
$nG=nH\cap G$ for every $n\in\mathbb{Z}$.

######  Notation 6.2.

i) Let $G$ be an abelian torsion free group and $Y$ be a subset of $G$. By
$\mathrm{PC}_{G}(Y)$ we mean the minimal pure subgroup of $G$ which includes
$Y.$

ii) The notation $\mathbb{Q}_{R}$ stands for the field of fractions of the
integral domain $R$.

######  Lemma 6.3.

Suppose $\mathbf{f}$ is an endomorphism of $M$ such that:

  1. $(a)\quad R$ is a commutative torsion free ring with $1_{R}$,

  2. $(b)\quad\mathbf{f}$ maps $\varphi_{0}(M)$ onto $\varphi_{0}(M)$,

  3. $(c)\quad M$ is a torsion free $R$-module,

  4. $(d)\quad\varphi_{\varepsilon}(x)\in\mathscr{L}^{ep}_{\infty,\aleph_{0}}(\tau_{R})$, for $\varepsilon\leq\varepsilon_{*}$ where $ep$ stands for existential positive (or

just generated from the atomic formulas by $\exists$ and $\wedge$),

  5. $(e)\quad\langle\varphi_{\varepsilon}(M):\varepsilon\leq\varepsilon_{*}\rangle$ is $\subseteq$-decreasing continuous of sub-modules of $M$,

  6. $(f)\quad\varphi_{\varepsilon_{*}}(M)=\\{0_{M}\\}$,

  7. $(g)\quad\varphi_{\varepsilon}(M)/\varphi_{\varepsilon+1}(M)$ is torsion free of rank 1, so isomorphic to some sub-$R$-module of $\mathbb{Q}_{R}$ considered as an $R$-module,

  8. $(h)\quad x_{\varepsilon}\in\varphi_{\varepsilon}(M)\backslash\varphi_{\varepsilon+1}(M)$ for $\varepsilon<\varepsilon_{*}$,

  9. $(i)\quad\varphi_{0}(M)=\mathrm{PC}_{M}(\\{x_{\varepsilon}:\varepsilon<\varepsilon_{*}\\})$.

Then $\mathbf{f}\restriction\varphi_{0}(M)$ is one-to-one.

###### Proof.

For $\varepsilon<\varepsilon_{*}$ let
$y_{\varepsilon}:=\mathbf{f}(x_{\varepsilon})$. As
$\varphi_{\varepsilon}(x)\in\mathscr{L}_{\infty,\aleph_{0}}(\tau_{R})$ is
existential positive, clearly $\mathbf{f}$ maps $\varphi_{\varepsilon}(M)$
into $\varphi_{\varepsilon}(M)$, hence
$y_{\varepsilon}\in\varphi_{\varepsilon}(M)$. Set

$\mathscr{U}:=\\{\varepsilon<\varepsilon_{*}:y_{\varepsilon}\in\varphi_{\varepsilon+1}(M)\\}.$

######  Claim 6.4.

$\mathscr{U}=\emptyset$.

###### Proof.

Assume towards a contradiction that $\mathscr{U}\neq\emptyset$ and let $\zeta$
be the first member of $\mathscr{U}$. As we are assuming “$\mathbf{f}$ maps
$\varphi_{0}(M)$ onto $\varphi_{0}(M)$” there is $z\in\varphi_{0}(M)$ such
that $\mathbf{f}(z)=x_{\zeta}$. Let $R^{+}:=R\setminus\\{0\\}$. Since
$z\in\varphi_{0}(M)$ by items (i) and (c) of the assumption of the claim we
can find

  1. $\bullet$

$b\in R^{+},n\in\mathbb{N}$

  2. $\bullet$

$\varepsilon_{0}<\ldots<\varepsilon_{n}<\varepsilon_{*}$ and

  3. $\bullet$

$a_{0},\dotsc,a_{n}\in R^{+}$,

such that

$M\models``bz=\sum\limits_{\ell\leq n}a_{\ell}x_{\varepsilon_{\ell}}".$

Now applying $\mathbf{f}$ we have

$M\models``bx_{\zeta}=\sum\limits_{\ell\leq
n}a_{\ell}y_{\varepsilon_{\ell}}".$

We consider two cases:

Case 1: $\varepsilon_{0}<\zeta$.

To see this, note that
$bx_{\zeta}\in\varphi_{\zeta}(M)\subseteq\varphi_{\varepsilon_{0}+1}(M)$ and
for $\ell>0$ we have
$a_{\ell}y_{\varepsilon_{\ell}}\in\varphi_{\varepsilon_{\ell}}(M)\subseteq\varphi_{\varepsilon_{0}+1}(M)$,
so as $bx_{\zeta}=\sum\limits_{\ell\leq n}a_{\ell}y_{\varepsilon_{\ell}}$ we
get $a_{0}y_{\varepsilon_{0}}\in\varphi_{\varepsilon_{0}+1}(M)$ and as
$a_{0}\in R^{+}$ by clause (f) we get
$y_{\varepsilon_{0}}\in\varphi_{\varepsilon_{0}+1}(M)$. This contradicts
$\varepsilon_{0}<\zeta=\min(\mathscr{U})$.

Case 2: $\varepsilon_{0}\geq\zeta$.

On the one hand as $b\in R^{+}$ clearly
$bx_{\zeta}\notin\varphi_{\zeta+1}(M)$. On the other hand

  1. $(2.1)$

$\varepsilon_{\ell}>\zeta\Rightarrow
a_{\ell}y_{\varepsilon_{\ell}}\in\varphi_{\varepsilon_{\ell}}(M)\subseteq\varphi_{\varepsilon_{0}+1}(M)\subseteq\varphi_{\zeta+1}(M)$
and

  2. $(2.2)$

$\varepsilon_{\ell}=\zeta\Rightarrow\ell=0\Rightarrow
y_{\varepsilon_{\ell}}=y_{\zeta}\in\varphi_{\zeta+1}(M)$.

Hence $\sum\limits_{\ell\leq
n}a_{\ell}y_{\varepsilon_{\ell}}\in\varphi_{\zeta+1}(M)$. Combining these
together we get a contradiction to

$M\models``bx_{\zeta}=\sum\limits_{\ell\leq
n}a_{\ell}y_{\varepsilon_{\ell}}",$

and we are done. The claim follows. ∎

Now, if $x\in\varphi_{0}(M)\backslash\\{0_{M}\\}$, then by clause (i) of the
assumption for some $b\in R^{+},n$ and
$\varepsilon_{0}<\ldots<\varepsilon_{n}<\varepsilon_{*}$ and $a_{\ell}\in
R^{+}$ for $\ell\leq n$ we have

$M\models``bx=\sum\limits_{\ell\leq n}a_{\ell}x_{\varepsilon_{\ell}}"$

hence

$b\mathbf{f}(x)=\sum\limits_{\ell\leq
n}a_{\ell}\mathbf{f}(x_{\alpha_{\ell}})\in
a_{0}y_{\varepsilon_{0}}+\varphi_{\varepsilon_{0}+1}(M).$

As $a_{0}\in R^{+},$ and in the light of clause (h) we observe
$y_{\varepsilon_{0}}\in\varphi_{\varepsilon_{0}}(M_{\varepsilon_{0}})\backslash\varphi_{\varepsilon_{0}+1}(M)$.
By items (f) and (g) of the assumption,
$a_{0}y_{\varepsilon_{0}}\notin\varphi_{\varepsilon_{0}+1}(M)$. Due to the
previous sentence, we know $b\mathbf{f}(x)\neq 0_{M}$. Hence
$\mathbf{f}(x)\neq 0_{M}$. So, we have proved that $\mathbf{f}$ maps any non-
zero member of $\varphi_{0}(M)$ into a non-zero member of $\varphi_{0}(M)$. In
other words, $\mathbf{f}\restriction\varphi_{0}(M)$ is one-to-one as promised.
∎

## References

  * [1] M. Asgharzadeh, M. Golshani, and Saharon Shelah, _Co-Hopfian and boundedly endo-rigid mixed abelian groups_ , arXiv: 2210.17210.
  * [2] Reinhold Baer,  Groups without proper isomorphic quotient groups.  Bull. Amer. Math. Soc. 50 (1944), 267-278.
  * [3] R. A. Beaumont and R. S. Pierce,  Partly transitive modules and modules with proper isomorphic submodules.  Trans. Amer. Math. Soc. 91 (1959), 209-219.
  * [4] R. A. Beaumont, Groups with isomorphic proper subgroups, Bull. Amer. Math. Soc. 51 (1945) 381-387.
  * [5] Jon Barwise, _Back and forth through infinitary logic_ , Studies in Model Theory, Math. Assoc. Amer., Buffalo, N.Y., 1973, pp. 5-34, MAA Studies in Math., Vol. 8.
  * [6] G. Braun and L. Strüngmann,  The independence of the notions of Hopfian and co-Hopfian abelian $p$-groups.  Proc. Amer. Math. Soc. 143 (2015), no. 8, 3331-3341.
  * [7] G. Baumslag, _Hopficity and Abelian Groups_ , in Topics in Abelian Groups (Proc. Sympos., New Mexico State Univ.), Scott Foresman and Co., Chicago, 1963. pp.331-335.
  * [8] A. L. S. Corner, _Three examples on hopficity in torsion-free abelian groups_ , Acta Math. Hungarica 16, No. 3-4(1965), 303-310.
  * [9] Peter Crawley,  An infinite primary abelian group without proper isomorphic subgroups ,  Bull. Amer. Math. Soc. 68 (1962), no. 5, 463-467.
  * [10] Paul C. Eklof and Alan Mekler, _Almost free modules: Set theoretic methods_ , North–Holland Mathematical Library, vol. 65, North–Holland Publishing Co., Amsterdam, 2002, Revised Edition.
  * [11] Erdos, Paul; Hajnal, Andras; Mate, Attila; Rado, Richard (1984), _Combinatorial set theory: partition relations for cardinals_ , Studies in Logic and the Foundations of Mathematics, vol. 106, Amsterdam: North-Holland Publishing Co.
  * [12] Paul C. Eklof and Saharon Shelah, _Absolutely rigid systems and absolutely indecomposable groups_ , abelian groups and modules (Dublin, 1998), Trends in Mathematics, Birkhäuser, Basel, 1999, math.LO/0010264, pp. 257–268.
  * [13] Laszlo Fuchs, _Abelian groups_ , Springer Monographs in Mathematics. Springer, Cham, 2015.
  * [14] László Fuchs, _Infinite abelian Groups_ , vol. I, II, Academic Press, New York, 1970, 1973.
  * [15] B. Goldsmith and K. Gong. _Algebraic entropies, Hopficity and co-Hopficity of direct sums of Abelian groups_ , Topol. Algebra Appl. 3 (2015), 75-85.
  * [16] Rüdiger Göbel and and Saharon Shelah, _Absolutely Indecomposable Modules, Proceedings of the American Mathematical Society_ , 135 (2007), 1641-1649.
  * [17] H. Hopf,_Fundamentalgruppe und zweite Bettische Gruppe_ , Comment. Math. Helv., 14 (1941), 257-309.
  * [18] I. Kaplansky, _A note on groups without isomorphic subgroups_ , Bull. Amer. Math. Soc. 51 (1945) 529-530.
  * [19] Marker, David Lectures on infinitary model theory. Lecture Notes in Logic, 46 Association for Symbolic Logic, Chicago, IL; Cambridge University Press, Cambridge, 2016.
  * [20] M.E. Nadel, Jonathan Stavi, _$L_{\infty\lambda}$ -equivalence, isomorphism and potential isomorphism_ Trans. Amer. Math. Soc. 236 (1978), 51-74.
  * [21] M.E. Nadel, _Scott heights of abelian groups_ , Journal of Symbolic Logic 59 (1994), 1351–1359.
  * [22] G. Paolini and S. Shelah, _On the existence of uncountable Hopfian and co-Hopfian abelian groups_ , to appear Israel J. Math. arXiv: 2107.11290.
  * [23] Jean-Pierre Serre, _How to use finite fields for problems concerning infinite fields_ , arXiv:0903.0517 [math.AG].
  * [24] W. Magnus, A. Karras and D. Solitar, _Combinatorial group theory_ , Interscience, New York, 1966.
  * [25] Saharon Shelah, _Infinite abelian groups, Whitehead problem and some constructions_ , Israel Journal of Mathematics 18 (1974), 243–256.
  * [26] Saharon Shelah, _Interpreting set theory in the endomorphism semi-group of afree algebra or in a category_ Ann. Sci. Univ. Clermont, (60 Math. No. 13), (1976). 1-29.
  * [27] Saharon Shelah, _Modules and infinitary logics, Groups and model theory_ , Contemp. Math., vol. 576, Amer. Math. Soc., Providence, RI, (20120) pp. 305-316.
  * [28] Saharon Shelah, _The lazy model theorist’s guide to stability_ , Logique et Analyse, 18 Anne, Vol. 71-72 (1975), 241-308.
  * [29] Saharon Shelah, _Better quasi-orders for uncountable cardinals_ , Israel J. Math. 42 (1982), no. 3, 177-226.
  * [30] W. Szmielew, _Elementary properties of abelian groups_ , Fundamenta Mathematica, 41 (1954), 203-271.
  * [31] Wolmer V. Vasconcelos, _On finitely generated flat modules_ Trans. Amer. Math. Soc. 138 (1969), 505?512.