# Naturality and Definability III

Mohsen Asgharzadeh  Mohsen Asgharzadeh, Hakimiyeh, Tehran, Iran.
<EMAIL_ADDRESS>,  Mohammad Golshani  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
Edmond J. Safra Campus, Givat Ram
The Hebrew University of Jerusalem
Jerusalem, 91904, Israel
and
Department of Mathematics
Hill Center - Busch Campus
Rutgers, The State University of New Jersey
110 Frelinghuysen Road
Piscataway, NJ 08854-8019 USA<EMAIL_ADDRESS>http://shelah.logic.at

###### Abstract.

In this paper, we deal with the notions of naturality from category theory and
definablity from model theory and their interactions. In this regard, we
present three results. First, we show, under some mild conditions, that
naturality implies definablity. Second, by using reverse Easton iteration of
Cohen forcing notions, we construct a transitive model of ZFC in which every
uniformisable construction is weakly natural. Finally, we show that if $F$ is
a natural construction on a class $\mathcal{K}$ of structures which is
represented by some formula, then it is uniformly definable without any extra
parameters. Our results answer some questions by Hodges and Shelah.

###### Key words and phrases:

Automorphisms of algebraic structures; definable functors; lifting morphisms;
forcing techniques; natural constructions; sorted modeles; uniformity.

###### 2010 Mathematics Subject Classification:

Primary 08A35; 03E35; 18A15.

The second author’s research has been supported by a grant from IPM (No.
1402030417). The third author would like to thank the Israel Science
Foundation (ISF) for partially supporting this research by grant No. 1838/19,
his research partially supported by the grant “Independent Theories” NSF-BSF
NSF 2051825, (BSF 3013005232). The third author is grateful to an individual
who prefers to remain anonymous for providing typing services that were used
during the work on the paper. This is publication 1245 of third author.

##  1\. Introduction

We are looking to find some interplay between the notions of naturality from
category theory and definability from set theory and model theory. This
continues the investigation initiated by Hodges and Shelah in [8] and [9].

In this paper, naturality is in the sense of Eilenberg and Mac Lane [3], i.e.,
for a class $\mathcal{K}$ of structures, the constriction $A\mapsto F(A)$, for
$A\in\mathcal{K}$, is natural, if for every $A\in\mathcal{K}$ and every
automorphism $a$ of $A$ there is an automorphism $f(a)$ of $F(A)$ such that
the following diagram is commutative

$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\scriptstyle{\subseteq}$$\textstyle{F(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f(a)}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\subseteq}$$\textstyle{F(A)}$

and the assignment $A\mapsto F(A)$ has the following two properties:

  1. (1)

$f({\rm id}_{A})={\rm id}_{F(A)}$, where ${\rm id}_{A}$ is the identity
function on $A$,

  2. (2)

for automorphisms $a,b$ of $A$, $f(ab)=f(a)f(b)$.

The challenging initial example for naturality was the dual of vector spaces,
see [3]. In [7, page 11] Lambek observed that divisible hulls of abelian
groups are not natural. Despite this, he pointed out that divisible hulls of
torsion-free abelian groups are natural, as it is left adjoint to the
inclusion functor.

Hodges and Shelah [8] observed that the construction of algebraic closure of
fields and the construction of divisible hulls are not natural. Also, they
showed that any left adjoint to any functor gives naturality. This may be
considered as a source to produce several natural constructions. According to
[1] in a category with injective hulls and a cogenerator, the injective hulls
is not natural, unless all objects are injective.

Naturality says that there is an splitting for ${\rm Aut}(F(A))\to{\rm
Aut}(A)$. From this one may define the notion of weak naturality, see [9,
Definition 4]. Hodges and Shelah [8] asked:

######  Question 1.1.

When does naturality imply definablity?

We use two-sorted models in order to deal with the question. This approach is
not new. For example, Harvey Friedman [4] already has considered some
constructions in algebra (e.g., direct product construction of pairs of
groups) as operations from relational structures of a certain fixed many
sorted relational type to structures of an enlarged many sorted relational
type. Also, Hodges and Shelah [9] interpreted Question 1.1 in terms of two-
sorted models. In sum, Question 1.1 can be discussed naturally in the context
of many sorted model theory.

In §2, and for the convenience of the readers, we review the concept of many
sorted models. According to its definition, any 2-sorted model
${\mathfrak{B}}$ gives us two groups $H:={\rm Aut}({\mathfrak{B}})$ and
$G:={\rm Aut}({\rm sort}_{1}({\mathfrak{B}}))$. If the restriction map
$\varphi:H\to G$ is well-defined and has a weak lifting $\psi$, we say
${\mathfrak{B}}$ gives us a uni-construction problem

${\mathbf{c}}=\langle{\mathfrak{B}},{\rm
sort}_{1}({\mathfrak{B}}),H,K,G,\varphi,\psi\rangle,$

where $K$ is the center of $H$ (see Definition 2.5). The new concept of uni-
construction problem will play the role of weak naturality. Then we introduce
the concept of solvability (see Definition 2.8), which plays the role of
definablity.

Concerning Question 1.1, in §3, we discuss some special kinds of 3-sorted
models, and prove the following as our first main result:

######  Theorem 1.2.

Let $\chi$ be a cardinal and let ${\mathbf{c}}$ be a uni-construction problem.
If ${\mathbf{c}}$ has no lifting, then in a forcing extension, ${\mathbf{c}}$
has no $\chi$-solution.

In [9, Theorem 10], Hodges and Shelah constructed a transitive model of ZFC in
which every uniformisable construction of a prescribed size is weakly natural.
Since their result has a cardinality restriction, they raised the following
natural conjecture.

######  Conjecture 1.3.

(See [9, Page 8]) The cardinality restriction can be removed.

In §4 we settle the conjecture, by iterating the main forcing construction of
[9, Theorem 10] using reverse Easton iteration of suitable Cohen forcing
notions. It may be nice to note that the use of forcing techniques in the
naturality problems comes back to Friedman [4] where he applied Easton product
Cohen-forcing, though the term naturalness in [4] is weaker than here.

Let $M$ be a model of set theory. Suppose some formula represents the
construction $F$ on the class $\mathcal{K}$ in $M$. If $F$ is natural and
there is only a set of isomorphism types of structures in $\mathcal{K}$, then
[8, Theorem 3] states that $F$ is definable in $M$ with parameters. In this
regard, Hodges and Shelah raised the following problem:

######  Problem 1.4.

(See [8, Problems (A) and (B)])

  1. (i)

Can the restriction “$\mathcal{K}$ contains only a set of isomorphism types”
be removed?

  2. (ii)

If $F$ has representing formula $\varphi$, is it always possible to define $F$
by a formula whose parameters are those in $\varphi$ and those needed to
define $\mathcal{K}$?

Recall that left adjoint defines a natural construction. It may be worth to
note that Hodges [6] used this and produced some natural constructions that
are set-theoretically definable from a parameter, via right Kan extension
along a small functor.

We want to restrict parameters from the defining formula of the natural
constructions. In §5 we apply techniques from many sorted models, and
reformulate the mentioned result of Hodges-Shelah [8, Theorem 3] about
naturality implies definablity. This reformulation helps us to control
parameters, and enables us to prove the following solution to Problem 1.4:

######  Theorem 1.5.

(Uniformity) Assume ${\mathfrak{B}}$ is a two sorted model in the vocabulary
$\tau$ and ${\mathfrak{A}}={\rm sort}_{1}({\mathfrak{B}})$. Suppose that
$\mathcal{K}$ is a class of $\tau$-models which is first order definable from
a parameter ${\mathbf{p}}$ such that for every ${\mathfrak{B}}\in\mathcal{K}$,
the natural homomorphism $\pi^{*}_{{\mathfrak{B}}}:{\rm
Aut}({\mathfrak{B}})\twoheadrightarrow{\rm Aut}({\rm
sort}_{1}({\mathfrak{B}}))$ splits. Then there exists a class function

$F:\\{{\rm
sort}_{1}({\mathfrak{B}}):{\mathfrak{B}}\in\mathcal{K}\\}\longrightarrow\mathcal{K},$

which is uniformly definable from the parameter ${\mathbf{p}}$ and ${\rm
sort}_{1}(F({\mathfrak{A}}))={\mathfrak{A}}$.

One may regard this uniformity result as an essential generalization of [8,
Theorem 3].

##  2\. The uni-construction problem

In this section we define the concepts of uni-construction problem, lifting,
weak lifting and solvability. We use many sorted models, to unify and simplify
our treatment.

Let us start by defining $n$-sorted model structures, where $n\geq 2$ is a
natural number. We only work with finite structures.

######  Definition 2.1.

An $n$-sorted model structure $\mathcal{M}$ is of the form

$\mathcal{M}=\big{(}\\{M_{1};\ldots;M_{n}\\};R_{1};\ldots;R_{m};f_{1};\ldots;f_{k};c_{1};\ldots;c_{l}\big{)}$

where

  1. (a)

the universes $M_{1};\ldots;M_{n}$ are nonempty,

  2. (b)

the relations $R_{1};\ldots;R_{m}$ are between elements of the universes. In
other words, for each $i$, there is some sequence $\langle
i_{1},\ldots,i_{s}\rangle$ from $\\{1,\ldots n\\}$ such that

$R_{i}\subseteq M_{i_{1}}\times\ldots\times M_{i_{s}},$

  3. (c)

the functions $f_{1};\ldots;f_{k}$ are between elements of the universes. In
other words, for each $i$, there is some sequence $\langle
i_{1},\ldots,i_{s},r\rangle$ from $\\{1,\ldots n\\}$ such that

$f_{i}:M_{i_{1}}\times\ldots\times M_{i_{s}}\to M_{r},$

  4. (d)

the distinguished constants $\\{c_{1};\ldots;c_{l}\\}$ are in the universes.

######  Notation 2.2.

Given an $n$-sorted model structure $\mathcal{M}$ as above, for any $1\leq
i\leq n,$ there is a natural model structure with the universe $M_{i}$, that
we denote by ${\rm sort}_{i}(\mathcal{M})$.

In what follow, we only work with the concepts of 2-sorted and 3-sorted model
structures.

######  Notation 2.3.

Let $H$ be a group which is not necessarily abelian.

  1. i)

By $\mathcal{Z}(H)$ we mean the center of $H$.

  2. ii)

Since $\mathcal{Z}(H)$ is a normal subgroup of $H$, it induces a group
structure on $H/\mathcal{Z}(H)$. By $\pi_{H}:H\rightarrow H/\mathcal{Z}(H)$ we
mean the natural group homomorphism from $H$ onto $H/\mathcal{Z}(H)$.

######  Definition 2.4.

Let $H$ and $G$ be two groups, which are not necessarily abelian. Assume
$\varphi:H\twoheadrightarrow G$ is an surjective homomorphism of groups.

  1. 1)

We say $\psi\in{\rm Hom}(G,H)$ splits $\varphi$, when $\varphi\circ\psi={\rm
id}_{G}$.

  2. 2)

We say $\psi$ weakly splits $\varphi$ provided:

    1. (a)

$\psi$ is a function from $G$ into $H$,

    2. (b)

$\varphi\circ\psi={\rm id}_{G}$,

    3. (c)

the composite mapping $\pi_{H}\circ\psi$ belongs to ${\rm
Hom}(G,H/\mathcal{Z}(H))$,

    4. (d)

$\psi(x^{-1})=(\psi(x))^{-1}$ and $\psi(1_{G})=1_{H}$.

  3. 3)

We say $\varphi$ splits (resp. weakly splits), if some $\psi$ splits (resp.
weakly splits) it.

We now define the concept of uni-construction problem, which plays a key role
in this paper.

######  Definition 2.5.

1) We say ${\mathbf{c}}$ is a uni-construction problem (${\rm ucp}$ in short),
when

${\mathbf{c}}=\langle{\mathfrak{B}}_{{\mathbf{c}}},{\mathfrak{A}}_{{\mathbf{c}}},H_{{\mathbf{c}}},K_{{\mathbf{c}}},G_{{\mathbf{c}}},\varphi_{{\mathbf{c}}},\psi_{{\mathbf{c}}}\rangle=\langle{\mathfrak{B}},{\mathfrak{A}},H,K,G,\varphi,\psi\rangle$

and it satisfies the following conditions:

  1. (a)

${\mathfrak{B}}$ is a two-sorted model,

  2. (b)

${\mathfrak{A}}:={\rm sort}_{1}({\mathfrak{B}}),$

  3. (c)

$H={\rm Aut}({\mathfrak{B}}),K=\mathcal{Z}(H)$ and $G={\rm
Aut}({\mathfrak{A}})$,

  4. (d)

$\varphi$ is the natural restriction map from $H$ into $G$, i.e.
$\varphi(f)=f{\restriction}{\mathfrak{A}}$, furthermore, $\varphi$ is a group
homomorphism:

$\textstyle{{\mathfrak{A}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\subseteq}$$\scriptstyle{f{\restriction}{\mathfrak{A}}}$$\textstyle{{\mathfrak{A}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\subseteq}$$\textstyle{{\mathfrak{B}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{f}}$$\textstyle{\mathfrak{B}}$

  5. (e)

$\varphi$ is onto $G$. So, for any $g:{\mathfrak{A}}\to{\mathfrak{A}}$ there
is an $f$ such that the following diagram commutes

$\textstyle{{\mathfrak{A}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\subseteq}$$\scriptstyle{g}$$\textstyle{{\mathfrak{A}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\subseteq}$$\textstyle{{\mathfrak{B}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\exists{f}}$$\textstyle{\mathfrak{B}}$

  6. (f)

$\psi$ weakly split $\varphi$.

2) We say ${\mathbf{c}}$ is a weak uni-constructive problem, if ${\mathbf{c}}$
is as above and it satisfies items (a)-(e) above.

######  Definition 2.6.

Let ${\mathbf{c}}$ be a uni-construction problem. Then the classes
$K^{1}_{{\mathbf{c}}}$ and $K^{2}_{{\mathbf{c}}}$ are defined as follows:

  1. (a)

$K^{1}_{{\mathbf{c}}}:=\\{{\mathfrak{A}}:{\mathfrak{A}}$ isomorphic to
${\mathfrak{A}}_{{\mathbf{c}}}\\}$,

  2. (b)

$K^{2}_{{\mathfrak{c}}}:=\\{{\mathfrak{B}}:{\mathfrak{B}}$ isomorphic to
${\mathfrak{B}}_{{\mathbf{c}}}\\}$.

######  Remark 2.7.

The assignment ${\rm sort}_{1}({\mathfrak{B}})\stackrel{{\scriptstyle
F}}{{\mapsto}}{\mathfrak{B}}$ on the domain of $K^{1}_{{\mathbf{c}}}$ is not
in general single-valued; but by clause (e) from Definition 2.5 it is single-
valued up to isomorphism over ${\mathfrak{B}}$. In particular, suppose
${\mathfrak{A}}\cong{\rm sort}_{1}({\mathfrak{B}})$. Then

$F({\rm sort}_{1}({\mathfrak{B}}))={\mathfrak{B}}\cong F({\mathfrak{A}}).$

######  Definition 2.8.

Let ${\mathbf{c}}$ be a uni-construction problem.

  1. (a)

We say ${\mathbf{c}}$ is solvable, when there is a class function
$F:K^{1}_{{\mathbf{c}}}\to K^{2}_{{\mathfrak{c}}}$, such that for each
${\mathfrak{B}}\in K^{2}_{{\mathfrak{c}}},$ $F({\rm
sort}_{1}({\mathfrak{B}}))={\mathfrak{B}}$.

  2. (b)

Let $F$ be as in clause (a). We say $F$ is definable, if there is a formula
$\theta$ such that

$F({\mathfrak{A}})={\mathfrak{B}}\Leftrightarrow\theta({\mathfrak{A}},{\mathfrak{B}}),$

for all two sorted models ${\mathfrak{B}}$ with ${\mathfrak{A}}:={\rm
sort}_{1}({\mathfrak{B}})$.

  3. (c)

We say ${\mathbf{c}}$ is purely solvable, when there is an $F$ as above which
is definable using only
$({\mathfrak{B}}_{{\mathbf{c}}},{\mathfrak{A}}_{{\mathbf{c}}})$ as a
parameter.

  4. (d)

We say ${\mathbf{c}}$ is $\chi$-solvable, when there is $F$ as above, which is
definable by some parameter $a\in{\mathscr{H}}(\chi)$.

Let us present with the following easy observation.

######  Observation 2.9.

Let ${\mathbf{c}}$ and ${\mathbf{d}}$ be two uni-construction problems,
satisfying the following conditions:

  1. (a)

${\mathfrak{A}}_{{\mathbf{c}}}={\mathfrak{A}}_{{\mathbf{d}}}$,

  2. (b)

${\mathfrak{B}}_{{\mathbf{d}}}$ expands ${\mathfrak{B}}_{{\mathbf{c}}}$.

If ${\mathbf{d}}$ is solvable, then ${\mathbf{c}}$ is solvable.

###### Proof.

Assume ${\mathbf{d}}$ is solvable, and let
$F_{{\mathbf{d}}}:K^{1}_{{\mathbf{d}}}\to K^{2}_{{\mathbf{d}}}$ witness it.
Let ${\mathfrak{A}}\in K^{1}_{{\mathbf{c}}}$. Since
$K^{1}_{{\mathbf{d}}}=K^{1}_{{\mathbf{c}}}$, we can define
$F_{{\mathbf{d}}}({\mathfrak{A}})$. Let $F_{{\mathbf{c}}}({\mathfrak{A}})$ be
the reduction of $F_{{\mathbf{d}}}({\mathfrak{A}})$ into the language of
${\mathfrak{B}}_{{\mathbf{c}}}$. Then
$F_{{\mathbf{c}}}:K^{1}_{{\mathbf{c}}}\to K^{2}_{{\mathbf{c}}}$ is well-
defined, and it witnesses that ${\mathbf{c}}$ is solvable. ∎

##  3\. From uni-construction to naturality

The main result of this section is Theorem 3.9, which shows that for a given
cardinal $\chi,$ if a uni-construction problem has no lifting, then it has no
$\chi$-solution in some forcing extension of the universe.

######  Hypothesis 3.1.

Let ${\mathfrak{C}}$ be a 3-sorted model such that
${\mathbf{c}}_{1,2},{\mathbf{c}}_{2,3},{\mathbf{c}}_{1,3}$ are weak uni-
construction problems, where:

  1. (1)

${\mathfrak{A}}_{{\mathbf{c}}_{1,2}}={\rm
sort}_{1}({\mathfrak{C}}),{\mathfrak{B}}_{{\mathbf{c}}_{1,2}}={\rm
sort}_{1,2}({\mathfrak{C}})$,

  2. (2)

${\mathfrak{A}}_{{\mathbf{c}}_{2,3}}={\rm
sort}_{1,2}({\mathfrak{C}}),{\mathfrak{B}}_{{\mathbf{c}}_{2,3}}={\mathfrak{C}}$,

  3. (3)

${\mathfrak{A}}_{{\mathbf{c}}_{1,3}}={\rm
sort}_{1}({\mathfrak{C}}),{\mathfrak{B}}_{{\mathbf{c}}_{1,3}}={\mathfrak{C}}$.

######  Lemma 3.2.

(Transitivity) Let ${\mathfrak{C}}$ be as in Hypothesis 3.1. If
${\mathbf{c}}_{1,2}$ and ${\mathbf{c}}_{2,3}$ are solvable, then
${\mathbf{c}}_{1,3}$ is solvable.

###### Proof.

According to Definition 2.6 there are definable class functions

  1. $F_{{\mathbf{c}}_{1,2}}:K^{1}_{{\mathbf{c}}_{1,2}}\to K^{2}_{{\mathbf{c}}_{1,2}}$, and

  2. $F_{{\mathbf{c}}_{2,3}}:K^{1}_{{\mathbf{c}}_{2,3}}\to K^{2}_{{\mathbf{c}}_{2,3}}$.

Recall from Hypothesis 3.1 that the following three equalities are satisfied

  1. (i)

$K^{1}_{{\mathbf{c}}_{1,2}}={\rm
sort}_{1}({\mathfrak{C}})=K^{1}_{{\mathbf{c}}_{1,3}}$,

  2. (ii)

$K^{1}_{{\mathbf{c}}_{2,3}}={\rm
sort}_{1,2}({\mathfrak{C}})=K^{2}_{{\mathbf{c}}_{1,2}}$,

  3. (iii)

$K^{2}_{{\mathbf{c}}_{2,3}}={\mathfrak{C}}=K^{2}_{{\mathbf{c}}_{1,3}}$.

This enables us to get the composition
$F_{{\mathbf{c}}_{1,3}}=F_{{\mathbf{c}}_{2,3}}\circ F_{{\mathbf{c}}_{1,2}}$.
Let us summarize things in the following commutative diagram:

---
$\textstyle{K^{1}_{{\mathbf{c}}_{1,3}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{{\mathbf{c}}_{1,3}}}$$\scriptstyle{=}$$\textstyle{K^{2}_{{\mathbf{c}}_{1,3}}}$$\textstyle{K^{1}_{{\mathbf{c}}_{1,2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{{\mathbf{c}}_{1,2}}}$$\textstyle{K^{2}_{{\mathbf{c}}_{2,3}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{K^{2}_{{\mathbf{c}}_{1,2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{K^{1}_{{\mathbf{c}}_{2,3}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{{\mathbf{c}}_{2,3}}}$

Then the function $F_{{\mathbf{c}}_{1,3}}$ witnesses that ${\mathbf{c}}_{1,3}$
is solvable. ∎

######  Conclusion 3.3.

Let ${\mathfrak{C}}$ be as in Hypothesis 3.1 and let $\chi$ be a cardinal. If
${\mathbf{c}}_{1,3}$ is not $\chi$-solvable and ${\mathbf{c}}_{2,3}$ is
solvable. Then ${\mathbf{c}}_{1,2}$ is not $\chi$-solvable.

###### Proof.

Assume by the way of contradiction that ${\mathbf{c}}_{1,2}$ is not
$\chi$-solvable. It follows from Lemma 3.2 that ${\mathbf{c}}_{1,3}$ is
$\chi$-solvable, a contradiction. ∎

######  Corollary 3.4.

Let ${\mathfrak{C}}$ be as in Hypothesis 3.1, and let $\chi$ be an infinite
cardinal. Assume in addition that

  1. (a)

$\varphi_{{\mathbf{c}}_{1,3}}$ has no weak lifting, and

  2. (b)

$\varphi_{{\mathbf{c}}_{2,3}}$ has a lifting.

Then there is a forcing extension of the universe in which
${\mathbf{c}}_{1,2}$ is not $\chi$-solvable.

###### Proof.

Fix an infinite cardinal $\chi$. As $\varphi_{{\mathbf{c}}_{2,3}}$ has a
lifting, and in the light of [8, Theorem 3] we observe that it is solvable.
Hence, for some cardinal $\chi^{\prime}\geq\chi,$ it is
$\chi^{\prime}$-solvable. As $\varphi_{{\mathbf{c}}_{1,3}}$ has no weak
lifting, thus by [9, Theorem 4], we can find a generic extension $V[G]$ of the
universe, in which $\varphi_{{\mathbf{c}}_{1,3}}$ is not
$\chi^{\prime}$-solvable. In view of Conclusion 3.3 we know that
${\mathbf{c}}_{1,2}$ is not $\chi^{\prime}$-solvable. Hence not
$\chi$-solvable in $V[G],$ as well. ∎

######  Definition 3.5.

Let $G^{\prime}$ be a group, and let $\psi$ be an automorphism of
$G^{\prime}$. Let $G$ be a $(\mathbb{Z},G^{\prime})$-construction. We are
going to use the multiplicative structure of $\mathbb{Z}$. In this regard, we
identify $\mathbb{Z}$ with $\\{y^{n}:n\in\mathbb{Z}\\}$. For simplicity, we
denote $(y^{n},x)$ with $y^{n}x$. So, as a set we have

$G:=\\{y^{n}x:n\in\mathbb{Z}\emph{ and }x\in G^{\prime}\\}.$

The multiplication of $x\in G^{\prime}$ from left hand side is defined via the
role

$x\times y:=y\psi(x)\in G.$

An easy induction shows that $x\times y^{n}:=y^{n}\psi^{n}(x)\in G$, and
consequently, $(G,\times)$ becomes a group, denoted by
$\mathbb{Z}\ltimes_{\psi}G^{\prime}$.

######  Proposition 3.6.

Let $G_{1}$ and $G_{2}$ be two groups and suppose there is an onto
homomorphism $\varphi_{1,2}:G_{2}\twoheadrightarrow G_{1}$ that has no lifting
but has a weak lifting. Then, there are $G_{3}$ and $\varphi_{2,3}$, such
that:

  1. (a)

$G_{3}$ is a group;

  2. (b)

$\varphi_{2,3}:G_{3}\twoheadrightarrow G_{2}$ is a surjective homomorphism
which has a lifting;

  3. (c)

$\varphi_{1,3}=\varphi_{1,2}\circ\varphi_{2,3}\in{\rm Hom}(G_{3},G_{2})$ is a
surjective homomorphism with no weak lifting.

###### Proof.

(a): First, we introduce the auxiliary group
$G^{\prime}_{3}:=\bigoplus\\{G_{2,n}:n\in{\mathbb{Z}}\\}$, where for each
$n\in{\mathbb{Z}},$ $G_{2,n}\cong G_{2}$. Let also $\psi_{n}$ denote the
corresponding isomorphism $\psi_{n}:G_{2}\to G_{2,n}$. Note that:

  1. (1)

$\bigcup\limits_{n\in{\mathbb{Z}}}G_{2,n}$ generates $G^{\prime}_{3}$,

  2. (2)

$G_{2,n}\cap\Sigma\\{G_{2,k}:k\in{\mathbb{Z}}\backslash\\{n\\}\\}=\\{1_{G^{\prime}_{3}}\\}$,
and

  3. (3)

the family of groups $\\{G_{2,n}:n\in{\mathbb{Z}}\\}$ pairwise commutes.

Let $\psi_{*}$ be the automorphism of $G^{\prime}_{3}$ such that, for each
$m,$

$\psi_{*}{\restriction}G_{2,m}=\psi_{m+1}\circ\psi^{-1}_{m}.$

Set $G_{3}:=\mathbb{Z}\ltimes_{\psi_{*}}G_{3}^{\prime}$. Recall from
Definition 3.5 that $G_{3}$ is generated by $G^{\prime}_{3}\cup\\{y\\}$,
subject to the following relations

$y^{-1}xy=\psi_{*}(x)\quad\forall x\in G^{\prime}_{3}.$

(b): Let $\varphi_{2,3}:G_{3}\rightarrow G_{2}$ be the unique homomorphism
from $G_{3}$ onto $G_{2}$ such that:

  1. $\bullet_{1}$

$\varphi_{2.3}{\restriction}G_{2,n}=\psi^{-1}_{n}\text{ for
}n\in{\mathbb{Z}}$,

  2. $\bullet_{2}$

$\varphi_{2.3}(y)=1_{G_{2}}$.

Now $\psi_{0}$ is clearly a lifting of $\varphi_{2,3}$, i.e. clause (b) holds.

$(c)$: In order to prove clause (c), we first prove the following claim.

######  Claim 3.7.

$G_{3}$ has trivial center.

###### Proof.

Suppose $x\in G^{\prime}_{3}\backslash\\{1_{G^{\prime}_{3}}\\}$. Clearly
$y^{-1}xy\neq x$ (using ${\rm supp}(x)$). Suppose $\alpha\in G_{3}\backslash
G^{\prime}_{3}$. By Definition 3.5, $\alpha$ has the form $y^{n}x_{1}$ with
$n\in{\mathbb{Z}},x_{1}\in G^{\prime}_{3}$ and $n\neq 0\vee x_{1}\neq
1_{G^{\prime}_{3}}$. Let $x_{2}\in
G^{\prime}_{2}\backslash\\{1_{G^{\prime}_{2}}\\}$ be such that

$\ell:=\min({\rm supp}(x_{2}))>\max({\rm supp}(x_{1}))+n.$

Suppose on the way of contradiction that $y^{n}x_{1}$ and $x_{2}$ commute with
each other. This yields that

$\begin{array}[]{ll}y^{n}(x_{1}x_{2})&=(y^{n}x_{1})x_{2}\\\
&=x_{2}(y^{n}x_{1})\\\ &=x_{2}(yy^{n-1}x_{1})\\\ &=(x_{2}y)y^{n-1}x_{1}\\\
&=(y\psi_{*}(x_{2}))y^{n-1}x_{1}\\\
&=y^{n}(\psi_{*}^{n}(x_{2})x_{1}).\end{array}$

By definition, this implies

$\psi_{*}^{n}(x_{2})x_{1}=x_{1}x_{2}\quad(+)$

Recall that $\psi_{*}{\restriction}G_{2,m}=\psi_{m+1}\circ\psi^{-1}_{m}$. By
applying this to $(+)$, we see some coordinate of its left hand side is
nonzero, but the corresponding coordinate in the right hand side is nonzero.
This contradiction shows that $G_{3}$ has trivial center, as claimed. ∎

By definition $\varphi_{1,3}\in{\rm Hom}(G_{3},G_{2})$ is onto. We show that
it has no weak lifting. Suppose towards contradiction that there is a function
$\psi:G_{1}\rightarrow G_{3}$ such that $\varphi_{1,3}\circ\psi={\rm
id}_{G_{1}}$ and the composite mapping $\pi_{H}\circ\psi$ belongs to ${\rm
Hom}(G_{1},G_{3}/\mathcal{Z}(G_{3}))$, where $\pi_{H}:G_{3}\rightarrow
G_{3}/\mathcal{Z}(G_{3})$ is the canonical homomorphism. But, according to
Claim 3.7, $G_{3}$ has trivial center. So $\psi$ is a homomorphism from
$G_{1}$ into $G_{3}$, hence $\varphi_{2,3}\circ\psi$ is a homomorphism from
$G_{1}$ into $G_{2}$; in fact, is one to one as $\varphi_{1,3}\circ\psi$ is.
This contradicts our assumption that $\varphi_{1,2}$ has no lifting. ∎

######  Corollary 3.8.

Let $\chi$ be a cardinal. Suppose $\varphi\in{\rm Hom}(G_{2},G_{1})$ is onto,
and does not split. Then there exists a weak uni-construction problem
${\mathbf{c}}$, such that in some forcing extension, ${\mathbf{c}}$ is not
$\chi$-solvable.

###### Proof.

Let $G_{1}$ and $G_{2}$ be as above and set $\varphi_{1,2}:=\varphi$. By
Proposition 3.6, we can find a group $G_{3}$ and two homomorphisms
$\varphi_{2,3}$ and $\varphi_{1,3}$, which fit in the following diagram

$\textstyle{G_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{1,2}}$$\textstyle{G_{1}}$$\textstyle{G_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{2,3}\rightsquigarrow\emph{
with no weak lifting}}$$\scriptstyle{\emph{with lifting
}\rightsquigarrow\varphi_{1,3}}$

Without loss of generality the groups $G_{1},G_{2}$ and $G_{3}$ are pairwise
disjoint. We define the three sorted model ${\mathfrak{C}}$ as follows:

  1. $(*)$

    1. (a)

the set of elements of ${\rm sort}_{\ell}({\mathfrak{C}})$ is the set of
elements of $G_{\ell}$ for $\ell=1,2,3$;

    2. (b)

$F^{{\mathfrak{C}}}_{\ell}=\varphi_{\ell,\ell+1}$ for $\ell=1,2$;

    3. (c)

for $\ell=1,2,3$ and $a\in G_{\ell}$, the homomorphism
$F^{{\mathfrak{C}}}_{\ell,a}:G_{\ell}\rightarrow G_{\ell}$ is defined as
$F_{\ell,a}^{{\mathfrak{C}}}(b)=a\cdot b$ for $b\in G_{\ell}$.

Now, we show that the assumptions of Corollary 3.4 hold. To this end, note
that we consider the sorts only as a set of elements and forget about the
multiplication in $G_{i}$. The entire structure is given by the functions. We
claim that

  1. (i)

$G_{1}\cong{\rm Aut}({\rm sort}_{1}({\mathfrak{C}})),G_{2}\cong{\rm Aut}({\rm
sort}_{1,2}({\mathfrak{C}}))$ and $G_{3}\cong{\rm Aut}({\mathfrak{C}})$ and

  2. (ii)

$\varphi_{i,j}:{\rm Aut}({\rm sort}_{i,j}({\mathfrak{C}}))\to{\rm Aut}({\rm
sort}_{i}({\mathfrak{C}}))$ is the natural restriction map.

For clause (i), we only show that $G_{3}\cong{\rm Aut}({\mathfrak{C}})$, as
the other cases can be proved in a similar way.

In order to see $G_{3}\cong{\rm Aut}({\mathfrak{C}})$, define the map

$\theta:G_{3}\rightarrow{\rm Aut}({\mathfrak{C}})$

which sends some element $c\in G_{3}$ to the map
$\sigma_{c}:{\mathfrak{C}}\to{\mathfrak{C}}$, which is defined via

$\sigma_{c}(g)=\left\\{\begin{array}[]{ll}c\cdot g&\mbox{if }g\in G_{3}\\\
\varphi_{2,3}(c)&\mbox{if }g\in G_{2}\\\ \varphi_{1,3}(c)&\mbox{if }g\in
G_{1}\end{array}\right.$

We have to show that $\theta$ is well-defined. Let $c\in G_{3}$. First we show
that $\theta(c)$ is a homomorphism of ${\mathfrak{C}}$. Consider $a\in G_{3}$,
and show it behaves well with respect to $F^{{\mathfrak{C}}}_{a,3}$:

$\sigma_{c}(F^{{\mathfrak{C}}}_{a,3}(b))=c\cdot b\cdot a=\sigma_{c}(b)\cdot
a=F^{{\mathfrak{C}}}_{a,3}(\sigma_{c}(b)),$

where, $b\in G_{3}$. Furthermore, it behaves well with respect to
$F_{2,3}^{{\mathfrak{C}}}$:

$\sigma_{c}(F_{2,3}^{{\mathfrak{C}}}(a))=\sigma_{c}(\varphi_{2,3}(a))=\varphi_{2,3}(c)\cdot\varphi_{2,3}(a)=\varphi_{2,3}(c\cdot
a)=\varphi_{2,3}(\sigma_{c}(a))=F_{2,3}^{{\mathfrak{C}}}(\sigma_{c}(a)).$

Similar arguments apply for $a,b\in G_{i}$ with $i=1,2,$ whence $\sigma_{c}$
is indeed a homomorphism of ${\mathfrak{C}}$. It is easy to see that each
$\theta(c)$ is a bijection, as it corresponds to left translation in the
groups. Thus $\theta(c)\in{\rm Aut}({\mathfrak{C}})$, and $\theta$ is well-
defined.

Now it remains to show that $\theta$ is an isomorphism of groups. In order to
be an homomorphism, it has to respect the function symbols. Let
$c_{1},c_{2}\in G_{3}$ be arbitrary. Then

$\theta(c_{1}\cdot c_{2})(a)=c_{1}\cdot c_{2}\cdot a=\theta(c_{1})(c_{2}\cdot
a)=(\theta(c_{1})\theta(c_{2}))(a),$

for any $a\in G_{3}$ and similarly for $a\in G_{i}$ with $i=1,2$.

Clearly, $\theta(c^{-1})=\theta(c)^{-1}$. Finally, the map $\theta$ is
injective, as for $c\neq c^{\prime}$ and $a\in G_{3}$, we have that

$\sigma_{c}(a)=c\cdot a\neq c^{\prime}\cdot a=\sigma_{c^{\prime}}(a).$

For surjectivity, consider an arbitrary automorphism $\sigma\in{\rm
Aut}({\mathfrak{C}})$. It is easy to check that for $c=\sigma(1_{G_{3}})$, we
have $\sigma=\sigma_{c}$, by using the identity

$\sigma(F^{{\mathfrak{C}}}_{a}(1_{G_{3}}))=F_{a}^{{\mathfrak{C}}}(\sigma(1_{G_{3}})).$

This concludes that $\theta:G_{3}\to{\rm Aut}({\mathfrak{C}})$ is an
isomorphism of groups.

To prove clause (ii), we have to show that the maps $\varphi_{i,j}$ correspond
to the natural restriction maps between the automorphism groups. This is
immediate, as we saw that any automorphism of ${\mathfrak{C}}$ corresponds to
the right translation on each of the sorts and if $\sigma$ restricted to the
$j^{th}$ sort is the translation by $b\in G_{j}$, then it is translation by
$\varphi_{i,j}(b)$ on the sort $i$.

In the light of Proposition 3.6, and in view of the way we defined the
3-sorted model ${\mathfrak{C}}$, we observe that
$\varphi_{{\mathbf{c}}_{1,3}}$ has no weak lifting, and
$\varphi_{{\mathbf{c}}_{2,3}}$ has a lifting. This allows us to apply
Corollary 3.4 and deduce that ${\mathbf{c}}_{1,2}$ is not $\chi$-solvable in
some forcing extension. But ${\mathbf{c}}_{1,2}={\mathbf{c}}$, and the
corollary follows. ∎

Now, we are ready to prove the main result of this section.

######  Theorem 3.9.

Let $\chi$ be a cardinal and ${\mathbf{c}}$ be a uni-construction problem. If
${\mathbf{c}}$ has no lifting, then in some forcing extension, ${\mathbf{c}}$
has no $\chi$-solution.

###### Proof.

Recall that $G_{{\mathbf{c}}}:={\rm Aut}({\mathfrak{A}}_{{\mathbf{c}}})$ and
$H_{{\mathbf{c}}}:={\rm Aut}({\mathfrak{B}}_{{\mathbf{c}}})$. We set
$G_{1}:=G_{{\mathbf{c}}},G_{2}:=H_{{\mathbf{c}}}$ and
$\varphi_{1,2}:=\varphi_{{\mathbf{c}}}$. In the light of Proposition 3.6 we
can find a group $G_{3}$, an a surjective homomorphism
$\varphi_{2,3}:G_{3}\twoheadrightarrow G_{2}$ which has a lifting such that
the homomorphism $\varphi_{1,3}=\varphi_{1,2}\circ\varphi_{2,3}\in{\rm
Hom}(G_{3},G_{2})$ has no weak lifting. Without loss of generality we may
assume that

$a\in{\mathfrak{B}}_{{\mathbf{c}}}\wedge b\in G_{3}\Rightarrow a\neq b.$

Let $\langle a_{\alpha}:\alpha<\alpha_{*}\rangle$ list the elements of
${\mathfrak{B}}_{{\mathbf{c}}}$. We define a 3-sorted model ${\mathfrak{C}}$
as follows:

  1. $(\ast)$

    1. (a)

${\rm sort}_{1,2}({\mathfrak{C}})={\mathfrak{B}}_{{\mathbf{c}}}$ so ${\rm
sort}_{1}({\mathfrak{C}})={\mathfrak{A}}_{{\mathbf{c}}}$;

    2. (b)

the set of elements of ${\rm sort}_{3}({\mathfrak{C}})$ is the set of elements
of $G_{3}$;

    3. (c)

$F^{{\mathfrak{C}}}_{1,\alpha}:G_{3}\rightarrow{\rm sort}_{1}({\mathfrak{C}})$
is defined by the help of $\varphi_{1,3}$. More precisely, suppose $b\in
G_{3}$, then we set
$F^{{\mathfrak{C}}}_{1,\alpha}(b)=(\varphi_{1,3}(b))(a_{\alpha})$;

    4. (d)

$F^{{\mathfrak{C}}}_{2,\alpha}:G_{3}\rightarrow{\mathfrak{B}}_{{\mathbf{c}}}$
is defined by:

$b\in G_{3}\Rightarrow
F^{{\mathfrak{C}}}_{2,\alpha}(b)=(\varphi_{2,3}(b))(a_{\alpha});$

    5. (e)

$F^{{\mathfrak{C}}}_{3,c}:G_{3}\rightarrow G_{3}$, where $c\in G_{3}$, is
defined by:

$b\in G_{3}\Rightarrow F^{{\mathfrak{C}}}_{3,c}(b)=cb.$

By the same vein as in the proof of Corollary 3.8, we observe that

  1. $(\ast\ast)$

    1. (a)

${\mathfrak{A}}_{{\mathbf{c}}_{1,2}}={\rm
sort}_{1}({\mathfrak{C}}),{\mathfrak{B}}_{{\mathbf{c}}_{1,2}}={\rm
sort}_{1,2}({\mathfrak{C}})$,

    2. (b)

${\mathfrak{A}}_{{\mathbf{c}}_{2,3}}={\rm
sort}_{1,2}({\mathfrak{C}}),{\mathfrak{B}}_{{\mathbf{c}}_{2,3}}={\mathfrak{C}}$,

    3. (c)

${\mathfrak{A}}_{{\mathbf{c}}_{1,3}}={\rm
sort}_{1}({\mathfrak{C}}),{\mathfrak{B}}_{{\mathbf{c}}_{1,3}}={\mathfrak{C}}$.

By our construction, it is easily seen tha $\varphi_{{\mathbf{c}}_{1,3}}$ has
no weak lifting, and $\varphi_{{\mathbf{c}}_{2,3}}$ has a lifting.

It follows from Corollary 3.4 that there exists a forcing extension $V[G]$ of
the universe in which ${\mathbf{c}}_{1,2}$ has no $\chi$-solution. But
${\mathbf{c}}_{1,2}={\mathbf{c}},$ and hence ${\mathbf{c}}$ has no
$\chi$-solution in $V[G]$. The theorem follows. ∎

##  4\. A global consistency result

For an infinite cardinal $\lambda$, let $\mathbb{S}_{\lambda}$ be the forcing
notion

$\mathbb{S}_{\lambda}=\\{p:\lambda^{++}\times\lambda^{++}\times\lambda^{++}\rightarrow
2:|p|\leq\lambda\\},$

ordered by reverse inclusion. Thus $\mathbb{S}_{\lambda}$ is forcing
equivalent to $\text{Add}(\lambda^{+},\lambda^{++})$, the Cohen forcing for
adding $\lambda^{++}$-many Cohen subsets of $\lambda^{+}.$ Furthermore, it is
$\lambda^{+}$-closed and satisfies the $\lambda^{++}$-c.c. In [9], the
following is proved.

######  Theorem 4.1.

(GCH) Let $M$ be an inner model of $\text{ZFC}+\text{GCH}$ and let $\lambda$
be an infinite cardinal of $M$. Let $\mathbb{Q}$ be the forcing notion
$\mathbb{S}_{\lambda}$ as computed in $M$ and let $\mathbb{G}$ be
$\mathbb{Q}$-generic over $M$. Then the following holds in $M[\mathbb{G}]:$

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

suppose $\mathbb{c}$ is a uniformisable uni-construction problem, such that
$\mathbb{c}$ is defined using parameters from $V$,
$\mathcal{B}_{\mathbb{c}}\in V$ and $\mathcal{B}_{\mathbb{c}}$ and ${\rm
Aut}(\mathcal{B}_{\mathbb{c}})$ have size $\leq\lambda.$ Then $\mathbb{c}$ is
weakly natural.

In this section we are going to prove a global version of this theorem, which
removes both the cardinality assumption and the parameter assumption from the
above result. The proof uses the reverse Easton iteration of forcing notions,
where we refer to [2] and [10, Chapter 21] for more details on this subject.

######  Theorem 4.2.

(GCH) Then there exists a GCH and cofinality preserving class generic
extension $V[\mathbb{G}]$ of the universe in which the following hold:

  1. $(*)$

If $\mathbb{c}$ is a uniformisable uni-construction problem, then $\mathbb{c}$
is weakly natural.

###### Proof.

Given an infinite cardinal $\lambda$ set $\mathbb{S}_{\lambda}$ be defined as
above. Let

$\mathbb{P}=\big{\langle}\langle\mathbb{P}_{\lambda}:\lambda\in{\rm
Ord}\rangle,\langle\mathchoice{\oalign{$\displaystyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{\lambda}:\lambda\in{\rm Ord}\rangle\big{\rangle}$

be the reverse Easton iteration of forcing notions, such that for each ordinal
$\lambda,\mathchoice{\oalign{$\displaystyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{\lambda}$ is forced to be the trivial forcing notion except
$\lambda$ is an infinite cardinal, in which case we let

$\Vdash_{\mathbb{P}_{\lambda}}$“$\mathchoice{\oalign{$\displaystyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{\lambda}=\mathchoice{\oalign{$\displaystyle\mathbb{S}$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle\mathbb{S}$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle\mathbb{S}$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle\mathbb{S}$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{\lambda}$”.

Thus a condition in $\mathbb{P}$ is a partial function $p$ such that:

  1. (1)

${\rm dom}(p)$ is a set of ordinal,

  2. (2)

if $\lambda\in{\rm dom}(p),$ then
$p\upharpoonright\lambda\Vdash_{\mathbb{P}_{\lambda}}$“$p(\lambda)\in\mathchoice{\oalign{$\displaystyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{\lambda}$”,

  3. (3)

for any regular cardinal $\kappa$, $|{\rm supp}(p)\cap\kappa|<\kappa$, where

${\rm supp}(p)=\\{\lambda\in{\rm
dom}(p):p\upharpoonright\lambda\Vdash_{\mathbb{P}_{\lambda}}\text{``}p(\lambda)\neq
1_{\mathchoice{\oalign{$\displaystyle\mathbb{Q}$\crcr\vbox
to0.60275pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle\mathbb{Q}$\crcr\vbox
to0.60275pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle\mathbb{Q}$\crcr\vbox
to0.60275pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle\mathbb{Q}$\crcr\vbox
to0.60275pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{\lambda}}\text{''}\\}.$

Let also

$\mathbb{G}=\big{\langle}\langle\mathbb{G}_{\lambda}:\lambda\in{\rm
Ord}\rangle,\langle\mathbb{H}_{\lambda}:\lambda\in{\rm
Ord}\rangle\big{\rangle}$

be $\mathbb{P}$-generic over $V$. Thus for each infinite cardinal $\lambda$,
$\mathbb{G}_{\lambda}=\mathbb{P}_{\lambda}\cap\mathbb{G}$ is
$\mathbb{P}_{\lambda}$-generic over $V$ and $H_{\lambda}$ is
$\mathchoice{\oalign{$\displaystyle\mathbb{S}$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle\mathbb{S}$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle\mathbb{S}$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle\mathbb{S}$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{\lambda}[\mathbb{G}_{\lambda}]$-generic over
$V[\mathbb{G}_{\lambda}]$.

We show that $V[\mathbb{G}]$ is as required. To make things easire, we recall
the following well-known result from [2] (also see [10, Chapter 21]):

######  Lemma 4.3.

  1. (1)

$V[\mathbb{G}]$ is a GCH and cofinality preserving class generic extension of
$V$,

  2. (2)

$V[\mathbb{G}]$ and $V[\mathbb{G}_{\lambda}]$ contain the same
$\lambda$-sequences of ordinals.

Now suppose $\mathbb{c}$ is a uniformisable uni-construction problem. Let
$\lambda$ be a large enough cardinal such that $\mathcal{B}_{\mathbb{c}}$,
${\rm Aut}(\mathcal{B}_{\mathbb{c}})$ and the parameters occurring in the
definition of $\mathbb{c}$ and the formula uniformizing it are all in
$V[\mathbb{G}_{\lambda}]$, and $|\mathcal{B}_{\mathbb{c}}|,|{\rm
Aut}(\mathcal{B}_{\mathbb{c}})|\leq\lambda$.

By Theorem $\ref{local-forcing},$ applied to the model
$V[\mathbb{G}_{\lambda}]$ and the uni-construction problem $\mathbb{c},$ we
can conclude that

$V[\mathbb{G}_{\lambda}][H_{\lambda}]\models$“$\mathbb{c}$ is weakly natural”.

On the other hand, $V[\mathbb{G}]$ is a generic extension of
$V[\mathbb{G}_{\lambda}][H_{\lambda}]$ by a class forcing notion which adds no
new subsets to $\lambda^{+}.$ Thus

$V[\mathbb{G}]\models$“$\mathbb{c}$ is weakly natural”.

The theorem follows. ∎

##  5\. Uniformity

In this section we deal with Problem 1.4. Our main result is Theorem 5.1,
which can be considered as a generalization of [8, Theorem 3]. We state our
result in terms of two sorted models.

######  Theorem 5.1.

Assume ${\mathfrak{B}}$ is a two sorted model in the vocabulary $\tau$ and
${\mathfrak{A}}={\rm sort}_{1}({\mathfrak{B}})$. Suppose that $K$ is a class
of $\tau$-models which is first order definable from a parameter
${\mathbf{p}}$ such that for every ${\mathfrak{B}}\in K$, the natural
homomorphism $\pi^{*}_{{\mathfrak{B}}}$ from ${\rm Aut}({\mathfrak{B}})$ onto
${\rm Aut}({\rm sort}_{1}({\mathfrak{B}}))$ splits. Then there exists a class
function $F$, which is uniformly definable from the parameter ${\mathbf{p}}$,
such that:

  1. (a)

the domain of $F$ is $K_{1}=\\{{\rm
sort}_{1}({\mathfrak{B}}):{\mathfrak{B}}\in K\\}$,

  2. (b)

if ${\mathfrak{A}}\in K_{1}$, then $F({\mathfrak{A}})\in K$ and ${\rm
sort}_{1}(F({\mathfrak{A}}))={\mathfrak{A}}$.

###### Proof.

We prove the theorem in a sequence of claims. We first show that we can be
reduced to the case where $K$ has only one equivalence class. To this end,
define the class function ${\mathbf{H}}$, with ${\rm dom}({\mathbf{H}})=K$, by

${\mathbf{H}}({\mathfrak{B}}):=\bigg{\\{}{\mathfrak{B}}^{\prime}:{\mathfrak{B}}^{\prime}$
is a $\tau$-model isomorphic to ${\mathfrak{B}}$ with universe the cardinal
$\|{\mathfrak{B}}\|\bigg{\\}}$.

Clearly, ${\mathbf{H}}$ is definable from the parameter ${\mathbf{p}}$. For
every ${\mathbf{x}}\in{\rm Range}({\mathbf{H}})$, set

  1. (1)

$K_{{\mathbf{x}}}:=\\{{\mathfrak{B}}\in
K:{\mathbf{H}}({\mathfrak{B}})={\mathbf{x}}\\}$,

  2. (2)

$K_{{\mathbf{x}},1}:=\\{{\rm sort}_{1}({\mathfrak{B}}):{\mathfrak{B}}\in
K_{{\mathbf{x}}}\\}$.

We remark that $\langle K_{\mathbf{x}}:{\mathbf{x}}\in{\rm
Rang}({\mathbf{H}})\rangle$ is a partition of $K$, which is uniformly
definable using the parameter ${\mathbf{p}}$, so it suffices to uniformly deal
with $K_{{\mathbf{x}}}$ for each ${{\mathbf{x}}}\in{\rm Range}({\mathbf{H}})$.
Thus, let us fix some ${{\mathbf{x}}}\in{\rm Range}({\mathbf{H}})$. Here, we
consider to $\Upsilon$. It is the family of all pairs $({\mathfrak{B}},\psi)$
equipped with the following properties:

  1. $\Upsilon_{1}$)

${\mathfrak{B}}\in K$ is such that for some
${\mathfrak{B}}^{\prime}\in{\mathbf{x}}$ the function
$b\mapsto({\mathfrak{B}}^{\prime},b)$, for $b\in{\mathfrak{B}}^{\prime}$, is
an isomorphism from ${\mathfrak{B}}^{\prime}$ onto ${\mathfrak{B}}$.

  2. $\Upsilon_{2}$)

$\psi$ is a weak lifting for ${\mathfrak{B}}$.

Let us collect all of them with a new name:

  1. (3)

Let
${\mathbf{y}}_{{\mathbf{x}}}:=\big{\\{}({\mathfrak{B}},\psi):({\mathfrak{B}},\psi)\in\Upsilon\big{\\}}$.

To make things be easier, we bring a series of claims (see Claim 5.2-5.9):

######  Claim 5.2.

${\mathbf{y}}_{{\mathbf{x}}}$ is non-empty.

###### Proof.

By our hypothesis, for each ${\mathfrak{B}}\in K$, there is a weak lifting for
${\mathfrak{B}}$, and hence ${\mathbf{y}}_{{\mathbf{x}}}$ is non-empty, as
requested. ∎

Let $S={\mathbf{y}}_{{\mathbf{x}}},$ and for each $s\in S$ set
$({\mathfrak{B}}_{s},\psi_{s})=s$. It then follows that

${\mathbf{y}}_{{\mathbf{x}}}=\\{({\mathfrak{B}}_{s},\psi_{s}):s\in S\\}.$

For $s\in S$ set

  1. (4)

${\mathfrak{A}}_{s}:={\rm sort}_{1}({\mathfrak{B}}_{s}).$

By replacing each ${\mathfrak{B}}_{s}$ by ${\mathfrak{B}}_{s}\times\\{s\\}$ if
necessary, we may assume that the ${\mathfrak{B}}_{s}$’s are pairwise
disjoint. For $b\in\bigcup\limits_{s\in S}{\mathfrak{B}}_{s}$ let

$r(b)=$ the unique $s\in S$ such that $b\in{\mathfrak{B}}_{s}.$

######  Claim 5.3.

${\mathbf{x}}$ is definable from ${\mathbf{y}}_{{\mathbf{x}}}$.

###### Proof.

Recall that ${\mathbf{x}}$ is equal to

$\bigg{\\{}{\mathfrak{B}}^{\prime}:\exists({\mathfrak{B}},\psi)\in{\mathbf{y}}_{{\mathbf{x}}}\big{(}\emph{the
function }b\mapsto({\mathfrak{B}}^{\prime},b)\emph{ induces
}{\mathfrak{B}}^{\prime}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}{\mathfrak{B}}\big{)}\bigg{\\}},$

which concludes the result. ∎

We say the ${\mathfrak{A}}_{\bullet}$-family
$\big{\langle}h_{s,t}:{\mathfrak{A}}_{s}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}{\mathfrak{A}}_{t}\big{\rangle}_{s,t\in
S}$ of isomorphisms is commutative provided:

  1. $\bullet$

$h_{r,s}\circ h_{s,t}=h_{r,t}$ and

  2. $\bullet$

$h_{s,s}={\rm id}_{{\mathfrak{A}}_{s}}$,

where $r,s,t\in S$. So, the diagram

$\textstyle{{\mathfrak{A}}_{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{s,t}}$$\textstyle{{\mathfrak{A}}_{t}}$$\textstyle{{\mathfrak{A}}_{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{r,s}}$$\scriptstyle{h_{r,t}}$

is commutative.

  1. (5)

By ${\rm Aut}_{1}({\mathbf{y}}_{{\mathbf{x}}})$ we mean

$\big{\\{}\bar{h}=\langle h_{s,t}:s,t\in S\rangle:\langle h_{s,t}:s,t\in
S\rangle\emph{ is a commutaive }{\mathfrak{A}}_{\bullet}-family\big{\\}}.$

Similarly, we say the ${\mathfrak{B}}_{\bullet}$-family $\langle
f_{s,t}:{\mathfrak{B}}_{s}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}{\mathfrak{B}}_{t}\rangle_{s,t\in
S}$ of isomorphisms is commutative if

$f_{r,s}\circ f_{s,t}=f_{r,t}$ and $f_{s,s}={\rm id}_{{\mathfrak{B}}_{s}}$ for
all $r,s,t\in S$. So, the diagram

$\textstyle{{\mathfrak{B}}_{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{s,t}}$$\textstyle{{\mathfrak{B}}_{t}}$$\textstyle{{\mathfrak{B}}_{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{r,s}}$$\scriptstyle{f_{r,t}}$

is commutative.

  2. (6)

By ${\rm Aut}_{2}({\mathbf{y}}_{{\mathbf{x}}})$ we mean

$\big{\\{}\bar{f}=\langle f_{s,t}:s,t\in S\rangle:\langle f_{s,t}:s,t\in
S\rangle\emph{ is a commutaive }{\mathfrak{B}}_{\bullet}-family\big{\\}}.$

For ${\mathfrak{A}}\in K_{{\mathbf{x}},1}$ let

  1. (7)

${\rm iso}_{{\mathbf{y}}_{{\mathbf{x}}}}({\mathfrak{A}})$ be the following

$\\{\bar{\pi}:\bar{\pi}=\langle\pi_{s}:s\in S\rangle\emph{ such that
}\pi_{s}:{\mathfrak{A}}_{s}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}{\mathfrak{A}}\\}.$

Suppose $\bar{\pi}\in{\rm iso}_{{\mathbf{y}}_{{\mathbf{x}}}}({\mathfrak{A}})$
and $s,t\in S$. We set

  1. (8)

$h_{\bar{\pi},s,t}=\pi^{-1}_{t}\circ\pi_{s}\in{\rm
iso}({\mathfrak{A}}_{s},{\mathfrak{A}}_{t}),$

  2. (9)

$\bar{h}_{\bar{\pi}}=\langle h_{\bar{\pi},s,t}:s,t\in S\rangle.$

This property can be summarized by the subjoined diagram:

$\textstyle{{\mathfrak{A}}_{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{s}}$$\scriptstyle{h_{\bar{\pi},s,t}}$$\textstyle{{\mathfrak{A}}_{t}}$$\textstyle{{\mathfrak{A}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{-1}_{t}}$

The proof of next claim is evident.

######  Claim 5.4.

Suppose ${\mathfrak{A}}\in K_{{\mathbf{x}},1}$ and $\bar{\pi}\in{\rm
iso}_{{\mathbf{y}}_{{\mathbf{x}}}}({\mathfrak{A}})$. Then
$\bar{h}_{\bar{\pi}}\in{\rm Aut}_{1}({\mathbf{y}}_{{\mathbf{x}}}).$

For ${\mathfrak{A}}\in K_{{\mathbf{x}},1}$, suppose $\bar{\pi}\in{\rm
iso}_{\bf y_{{\mathbf{x}}}}({\mathfrak{A}})$ and $\bar{g}\in{\rm
Aut}_{2}({\mathbf{y}}_{{\mathbf{x}}})$ are such that
$g_{s,s}=\psi_{s}(h_{\bar{\pi},s,s})$ 111Recall that $\psi_{s}$ is a weak
lifting for ${\mathfrak{B}}_{s}$. and $h_{\bar{\pi},s,t}\subseteq g_{s,t}$ for
$s,t\in S$. Let also $\bar{b}=\langle b_{s}:s\in S\rangle$. We say the triple
$(\bar{\pi},\bar{g},\bar{b})$ is _match_ , if $\bar{\pi},$ $\bar{g}$ and
$\bar{b}$ are as above, $b_{s}\in{\mathfrak{B}}_{s}$ and
$g_{s,t}(b_{t})=b_{s}$ for $s,t\in S$.

Let $\Sigma({\mathbf{x}},{\mathfrak{A}})$ denote the family of all triples
$(\bar{\pi},\bar{g},\bar{b})$ as above which are match. The following diagram
summarizes the above situation:

$\textstyle{b_{s}\in{\mathfrak{B}}_{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{s,t}}$$\textstyle{{\mathfrak{B}}_{t}\ni
b_{t}}$$\textstyle{{\mathfrak{A}}_{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{s}}$$\scriptstyle{h_{\bar{\pi},s,t}}$$\scriptstyle{\subseteq}$$\textstyle{{\mathfrak{A}}_{t}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\subseteq}$$\textstyle{{\mathfrak{A}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{-1}_{t}}$

  1. (10)

$X_{{\mathbf{y}}_{{\mathbf{x}}}}({\mathfrak{A}}):=\big{\\{}(\bar{\pi},\bar{g},\bar{b}):(\bar{\pi},\bar{g},\bar{b})\in\Sigma({\mathbf{x}},{\mathfrak{A}})\big{\\}}$.

For ${\mathfrak{A}}\in K_{{\mathbf{x}},1}$, let $E_{{\mathfrak{A}}}$ be the
following two place relation on
$X_{{\mathbf{y}}_{{\mathbf{x}}}}({\mathfrak{A}})$:

$(\bar{\pi},\bar{g}_{1},\bar{b}_{1})E_{{\mathfrak{A}}}(\pi_{2},\bar{g}_{2},\bar{b}_{2})$

if and only if

  * •

$(\bar{\pi}_{1},\bar{g}_{1},\bar{b}_{1})$ and
$(\bar{\pi}_{2},\bar{g}_{2},\bar{b}_{2})$ belong to
$X_{{\mathbf{y}}_{{\mathbf{x}}}}({\mathfrak{A}})$,

  * •

letting $\widetilde{\pi}_{s}=\pi^{-1}_{1,s}\pi_{2,s}$ and
$h_{s}=\psi_{s}(\widetilde{\pi}_{s}^{-1})$, we have $h_{s}(b_{1,s})=b_{2,s}$.

Note that $\widetilde{\pi}_{s}$ is an automorphism of ${\mathfrak{A}}_{s}$,
hence so is $\widetilde{\pi}_{s}^{-1}$, and hence
$h_{s}=\psi_{s}(\widetilde{\pi}_{s}^{-1})$ is a member of $H_{s}={\rm
Aut}({\mathfrak{B}}_{s})$ which extends $\widetilde{\pi}_{s}$. Since
$b_{1,2},b_{2,s}\in{\mathfrak{B}}_{s}$ so
$h_{s}(b_{1,s})=b_{2,s}\in{\mathfrak{B}}_{s}$ is meaningful.

######  Claim 5.5.

$E_{{\mathfrak{A}}}$ is an equivalence relation.

###### Proof.

It is clearly reflexive and symmetric. To show that $E_{{\mathfrak{A}}}$ is
transitive, assume that

  1. (a)

$(\bar{\pi}_{1},\bar{g}_{1},\bar{b}_{1})E_{{\mathfrak{A}}}(\bar{\pi}_{2},\bar{g}_{2},\bar{b}_{2})$,

  2. (b)

$(\bar{\pi}_{2},\bar{g}_{2},\bar{b}_{2})E_{{\mathfrak{A}}}(\bar{\pi}_{3},\bar{g}_{3},\bar{b}_{3})$.

By the definition of $E_{{\mathfrak{A}}}$

  1. (c)

$(\psi_{s}(\pi^{-1}_{1,s}\pi_{2,s})^{-1})(b_{1,s})=b_{2,s}$ for $s\in S$

  2. (d)

$(\psi_{s}(\pi^{-1}_{2,s}\pi_{3,s})^{-1})(b_{2,s})=b_{3,s}$ for $s\in S$.

Hence

$\begin{array}[]{ll}b_{3,s}&=(\psi_{s}(\pi^{-1}_{2,s}\pi_{3,s})^{-1})(b_{2,s})\\\
&=(\psi_{s}(\pi^{-1}_{2,s}\pi_{3,s})^{-1})((\psi_{s}(\pi^{-1}_{1,s}\pi_{2,s})^{-1})(b_{1,s}))\\\
&=(\psi_{s}\big{(}(\pi^{-1}_{2,s}\pi_{3,s})^{-1}\circ(\pi^{-1}_{1,s}\pi_{2,s})^{-1}\big{)}(b_{1,s}))\\\
&=\psi_{s}((\pi^{-1}_{1,s}\pi_{3,s})^{-1})(b_{1,s}),\end{array}$

as requested. ∎

We define a model ${\mathfrak{B}}^{\prime}_{{\mathfrak{A}}}$ as follows:222For
simplicity, we assume that the structures ${\mathfrak{B}}$ are relational, so
they only contain relations.

  1. (11.1)

the universe of ${\mathfrak{B}}^{\prime}_{{\mathfrak{A}}}$ is
$X_{{\mathbf{y}}_{{\mathbf{x}}}}({\mathfrak{A}})$,

  2. (11.2)

if $R\in\tau$ is an $n$-place relation, then

$R^{{\mathfrak{B}}^{\prime}_{{\mathfrak{A}}}}:=\bigg{\\{}\big{\langle}(\bar{\pi}_{\ell},\bar{g}_{\ell},\bar{b}_{\ell}):\ell<n\big{\rangle}\in
X_{{\mathbf{y}}_{{\mathbf{x}}}}({\mathfrak{A}})^{n}:\exists s\in
S\big{(}\langle b_{\ell,s}:\ell<n\rangle\in
R^{{\mathfrak{B}}_{s}}\big{)}\bigg{\\}}.$

######  Claim 5.6.

In the definition of $R^{{\mathfrak{B}}^{\prime}_{{\mathfrak{A}}}}$, we can
replace “$\exists s\in S$” by “$\forall s\in S$”.

###### Proof.

Suppose $s,t\in S$ and
$\langle(\bar{\pi}_{\ell},\bar{g}_{\ell},\bar{b}_{\ell}):\ell<n\rangle\in
X_{{\mathbf{y}}_{{\mathbf{x}}}}({\mathfrak{A}})^{n}$. We have to show that

$\langle b_{\ell,s}:\ell<n\rangle\in
R^{{\mathfrak{B}}_{s}}\Leftrightarrow\langle b_{\ell,t}:\ell<n\rangle\in
R^{{\mathfrak{B}}_{t}}.$

This follows from the fact that
$g_{s,t}:{\mathfrak{B}}_{s}\rightarrow{\mathfrak{B}}_{t}$ is an isomorphism
and $g_{s,t}(b_{\ell,s})=b_{\ell,t}$. ∎

######  Claim 5.7.

Assume
$(\bar{\pi}^{\iota}_{\ell},\bar{g}^{\iota}_{\ell},\bar{b}^{\iota}_{\ell})\in
X_{{\mathbf{y}}_{{\mathbf{x}}}}({\mathfrak{A}})$, for $\ell<n,\iota=1,2$, and
suppose that for each $\ell<n,$
$(\bar{\pi}^{1}_{\ell},\bar{g}^{1}_{\ell},\bar{b}^{1}_{\ell})E_{{\mathfrak{A}}}(\bar{\pi}^{2}_{\ell},\bar{g}^{2}_{\ell},\bar{b}^{2}_{\ell}).$
Then

$\langle(\bar{\pi}^{1}_{\ell},\bar{g}^{1}_{\ell},\bar{b}^{1}_{\ell}):\ell<n\rangle\in
R^{{\mathfrak{B}}^{\prime}_{{\mathfrak{A}}}}\Leftrightarrow\langle(\bar{\pi}^{2}_{\ell},\bar{g}^{2}_{\ell},\bar{b}^{2}_{\ell}):\ell<n\rangle\in
R^{{\mathfrak{B}}^{\prime}_{{\mathfrak{A}}}}.$

###### Proof.

Assume
$\langle(\bar{\pi}^{1}_{\ell},\bar{g}^{1}_{\ell},\bar{b}^{1}_{\ell}):\ell<n\rangle\in
R^{{\mathfrak{B}}^{\prime}_{{\mathfrak{A}}}}.$ For each $\ell<n$, we have

$(\bar{\pi}^{1}_{\ell},\bar{g}^{1}_{\ell},\bar{b}^{1}_{\ell})E_{{\mathfrak{A}}}(\bar{\pi}^{2}_{\ell},\bar{g}^{2}_{\ell},\bar{b}^{2}_{\ell}).$

This in turns imply that

$\psi_{s}((\pi^{2}_{\ell,s})^{-1}\pi^{1}_{\ell,s})(b^{1}_{\ell,s})=b^{2}_{\ell,s},$

where $s\in S$. As $\psi_{s}((\pi^{2}_{\ell,s})^{-1}\pi^{1}_{\ell,s})$ is an
automorphism of ${\mathfrak{B}}_{s}$, we have

$\langle b^{1}_{\ell,s}:\ell<n\rangle\in
R^{{\mathfrak{B}}_{s}}\Leftrightarrow\langle b^{2}_{\ell,s}:\ell<n\rangle\in
R^{{\mathfrak{B}}_{s}}.$

Hence:

$\begin{array}[]{ll}\big{\langle}(\bar{\pi}^{1}_{\ell},\bar{g}^{1}_{\ell},\bar{b}^{1}_{\ell}):\ell<n\big{\rangle}\in
R^{{\mathfrak{B}}^{\prime}_{{\mathfrak{A}}}}&\Rightarrow\exists s\in S\emph{
such that }\langle b^{1}_{\ell,s}:\ell<n\rangle\in R^{{\mathfrak{B}}_{s}}\\\
&\Rightarrow\langle b^{2}_{\ell,s}:\ell<n\rangle\in R^{{\mathfrak{B}}_{s}}\\\
&\Rightarrow\big{\langle}(\bar{\pi}^{2}_{\ell},\bar{g}^{2}_{\ell},\bar{b}^{2}_{\ell}):\ell<n\big{\rangle}\in
R^{{\mathfrak{B}}^{\prime}_{{\mathfrak{A}}}}.\end{array}$

By symmetry

$\big{\langle}(\bar{\pi}^{2}_{\ell},\bar{g}^{2}_{\ell},\bar{b}^{2}_{\ell}):\ell<n\big{\rangle}\in
R^{{\mathfrak{B}}^{\prime}_{{\mathfrak{A}}}}\Rightarrow\big{\langle}(\bar{\pi}^{1}_{\ell},\bar{g}^{1}_{\ell},\bar{b}^{1}_{\ell}):\ell<n\big{\rangle}\in
R^{{\mathfrak{B}}^{\prime}_{{\mathfrak{A}}}}.$

The claim follows. ∎

It follows from the above claim that $E_{{\mathfrak{A}}}$ is a congruence
relation on ${\mathfrak{B}}^{\prime}_{{\mathfrak{A}}}$. Define a function
$\mathbb{k}$ with domain ${\mathfrak{A}}$ as follows:

$\mathbb{k}(a)=\\{(\bar{\pi},\bar{g},\bar{b})\in
X_{{\mathbf{y}}_{{\mathbf{x}}}}({\mathfrak{A}}):b_{s}=\pi_{s}^{-1}(a)\text{~{}for
all~{}}s\in S\\}.$

######  Claim 5.8.

If $a\in{\mathfrak{A}}$, then $\mathbb{k}(a)$ is an
$E_{{\mathfrak{A}}}$-equivalence class.

###### Proof.

Let us first show that it is closed under the relation $E_{{\mathfrak{A}}}$,
in the sense that if $(\bar{\pi}_{1},\bar{g}_{1},\bar{b}_{1})\in\mathbb{k}(a)$
and
$(\bar{\pi}_{1},\bar{g}_{1},\bar{b}_{1})E_{{\mathfrak{A}}}(\bar{\pi}_{2},\bar{g}_{2},\bar{b}_{2}),$
then $(\bar{\pi}_{2},\bar{g}_{2},\bar{b}_{2})\in\mathbb{k}(a)$. Thus suppose
$s\in S$. Then

  * •

$b_{1,s}=\pi_{1,s}^{-1}(a)$,

  * •

$\psi_{s}(\pi^{-1}_{2,s}\pi_{1,s})(b_{1,s})=b_{2,s}$.

It then follows that

$\begin{array}[]{ll}b_{2,s}&=\psi_{s}(\pi^{-1}_{2,s}\pi_{1,s})(b_{1,s})\\\
&=\psi_{s}(\pi^{-1}_{2,s}\pi_{1,s})(\pi_{1,s}^{-1}(a))\\\
&=\pi^{-1}_{2,s}\pi_{1,s}\pi_{1,s}^{-1}(a)\\\ &=\pi^{-1}_{2,s}(a).\end{array}$

Thus $(\bar{\pi}_{2},\bar{g}_{2},\bar{b}_{2})\in\mathbb{k}(a)$, as requested.

Next we show that if
$(\bar{\pi}_{1},\bar{g}_{1},\bar{b}_{1}),(\bar{\pi}_{2},\bar{g}_{2},\bar{b}_{2})\in\mathbb{k}(a)$,
then
$(\bar{\pi}_{1},\bar{g}_{1},\bar{b}_{1})E_{{\mathfrak{A}}}(\bar{\pi}_{2},\bar{g}_{2},\bar{b}_{2}).$
Let $s\in S$ and set $h_{s}=\psi_{s}(\pi^{-1}_{2,s}\pi_{1,s}).$ We have to
show that $h_{s}(b_{1,s})=b_{2,s}$. But we have

$\begin{array}[]{ll}h_{s}(b_{1,s})&=\psi_{s}(\pi^{-1}_{2,s}\pi_{1,s})(b_{1,s})\\\
&=\psi_{s}(\pi^{-1}_{2,s}\pi_{1,s})(\pi_{1,s}^{-1}(a))\\\
&=\pi^{-1}_{2,s}\pi_{1,s}\pi_{1,s}^{-1}(a)\\\ &=\pi^{-1}_{2,s}(a)\\\
&=b_{2,s}.\end{array}$

We are done. The claim follows immediately. ∎

######  Claim 5.9.

${\mathfrak{B}}^{\prime}_{{\mathfrak{A}}}/E_{{\mathfrak{A}}}$ is isomorphic to
${\mathfrak{B}}_{s},$ for $s\in S.$

###### Proof.

Fix $s\in S$, and let $\phi:{\mathfrak{A}}_{s}\simeq{\mathfrak{A}}$ be an
isomorphism. Set

$Y_{s,\phi}:=\\{(\bar{\pi},\bar{g},\bar{b})\in
X_{{\mathbf{y}}_{{\mathbf{x}}}}({\mathfrak{A}}):\pi_{s}=\phi\\},$

and define a function

$\rho_{s,\phi}:Y_{s,\phi}\rightarrow{\mathfrak{B}}_{s}$

as $\rho_{s,\phi}((\bar{\pi},\bar{g},\bar{b}))=b_{s}$. Clearly,
$\rho_{s,\phi}$ is well-defined. Next, we bring the following two auxiliary
observations:

  * $(*)_{5.9.1}$

Suppose $x_{1}=(\bar{\pi}_{1},\bar{g}_{1},\bar{b}_{1})$ and
$x_{2}=(\bar{\pi}_{2},\bar{g}_{2},\bar{b}_{2})$ are in $Y_{s,\phi}$. Then

$x_{1}E_{{\mathfrak{A}}}x_{2}\Rightarrow\rho_{s,\phi}(x_{1})=\rho_{s,\phi}(x_{2}).$

###### Proof..

Suppose that $x_{1}E_{{\mathfrak{A}}}x_{2}$. It then follows that
$\psi_{s}(\pi^{-1}_{2,s}\pi_{1,s})(b_{1,s})=b_{2,s}$. But as
$\pi_{1,s}=\phi=\pi_{2s}$, we have

$\psi_{s}(\pi^{-1}_{2,s}\pi_{1,s})(b_{1,s})=\psi_{s}({\rm
id}_{{\mathfrak{A}}_{s}})(b_{1,s})={\rm
id}_{{\mathfrak{B}}_{s}}(b_{1,s})=b_{1,s},$

and thus $\rho_{s,\phi}(x_{1})=b_{1,s}=b_{2,s}=\rho_{s,\phi}(x_{2})$. ∎

Suppose $\bar{\pi}_{2}=\langle\pi^{2}_{s}:s\in S\rangle\in{\rm
iso}_{{\mathbf{y}}_{{\mathbf{x}}}}({\mathfrak{A}})$ and let
$x_{1}=(\bar{\pi}_{1},\bar{g}_{1},\bar{b}_{1})\in
X_{{\mathbf{y}}_{{\mathbf{x}}}}({\mathfrak{A}})$. Recall that
$\widetilde{\pi}_{s}=\pi^{-1}_{1,s}\pi_{2,s}.$ For simplicity, we set:

  * •

$\widetilde{\Pi}_{s}:=\psi_{s}(\widetilde{\pi}_{s})\in{\rm
Aut}({\mathfrak{B}}_{s})$.

  * $(*)_{5.9.2}$

Suppose $\bar{\pi}_{*}=\langle\pi^{*}_{s}:s\in S\rangle\in{\rm
iso}_{{\mathbf{y}}_{{\mathbf{x}}}}({\mathfrak{A}})$ and $s(*)\in S$. Let
$x_{1}=(\bar{\pi}_{1},\bar{g}_{1},\bar{b}_{1})\in
X_{{\mathbf{y}}_{{\mathbf{x}}}}({\mathfrak{A}})$. Then there is
$x_{2}=(\bar{\pi}_{2},\bar{g}_{2},\bar{b}_{2})\in[x_{1}]_{E_{{\mathfrak{A}}}}$
such that

    * $\bullet$

$\bar{\pi}_{2}=\bar{\pi}_{*}$ and

    * $\bullet$

$b_{2,s(*)}=\widetilde{\Pi}_{s(*)}^{-1}b_{1,s(*)}$.

###### Proof..

In order to define $x_{2}$, we set

  * (i)

$\bar{\pi}_{2}=\bar{\pi}_{*}$,

  * (ii)

for $s,t\in S$, $g_{2,t,s}=\psi_{t}((\pi^{-1}_{1,t}\pi_{2,t})^{-1})\circ
g_{1,t,s}\circ\psi_{s}(\pi^{-1}_{1,s}\pi_{2,s})$.

The following commutative diagram summarizes the above situation:

---
$\textstyle{{\mathfrak{B}}_{t}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{2,t,s}}$$\textstyle{{\mathfrak{B}}_{s}}$$\textstyle{{\mathfrak{B}}_{t}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{\Pi}_{t}}$$\scriptstyle{g_{1,t,s}}$$\textstyle{{\mathfrak{B}}_{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{\Pi}_{s}}$$\textstyle{{\mathfrak{A}}_{t}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\subseteq}$$\textstyle{{\mathfrak{A}}_{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\subseteq}$$\textstyle{{\mathfrak{A}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{2,t}^{-1}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathfrak{A}}_{t}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{1,t,s}}$$\scriptstyle{\pi_{1,t}}$$\scriptstyle{\widetilde{\pi}_{t}}$$\scriptstyle{\subseteq}$$\textstyle{{\mathfrak{A}}_{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\subseteq}$$\scriptstyle{\pi_{1,s}}$$\scriptstyle{\widetilde{\pi}_{s}}$$\textstyle{{\mathfrak{A}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{2,s}^{-1}}$

Let us depict the resulting commutative diagram:

$\textstyle{{\mathfrak{B}}_{s}}$$\textstyle{{\mathfrak{B}}_{t}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{\Pi}_{t}^{-1}}$$\scriptstyle{g_{2,t,s}}$$\textstyle{{\mathfrak{B}}_{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{\Pi}_{s}}$$\textstyle{{\mathfrak{B}}_{t}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{1,t,s}}$

It is easily seen that $x_{2}$ as defined above is as required. Indeed, by
(i), $\bar{\pi}_{2}=\bar{\pi}_{*}$. Also, it is clear that for each $s\in S$,
$g_{2,s,s}={\rm id}_{{\mathfrak{B}}_{s}}$. Next, we show the following diagram

$\textstyle{{\mathfrak{B}}_{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{2,s,t}}$$\textstyle{{\mathfrak{B}}_{t}}$$\textstyle{{\mathfrak{B}}_{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{2,r,s}}$$\scriptstyle{g_{2,r,t}}$

is commutative where $r,s,t\in S$. To check this, we first recall that
$\widetilde{\pi}_{s}=\pi^{-1}_{1,s}\pi_{2,s}$, then we have

$\begin{array}[]{ll}g_{2,r,s}\circ
g_{2,s,t}&=\big{(}\psi_{r}(\widetilde{\pi}_{r})^{-1}\circ
g_{1,r,s}\circ\psi_{s}(\widetilde{\pi}_{s})\big{)}\circ\big{(}\psi_{s}(\widetilde{\pi}_{s})^{-1}\circ
g_{1,s,t}\circ\psi_{t}(\widetilde{\pi}_{t})\big{)}\\\
&=\psi_{r}(\pi^{-1}_{1,r}\pi_{2,r})^{-1}\circ g_{1,r,s}\circ
g_{1,s,t}\circ\psi_{t}(\pi^{-1}_{1,t}\pi_{2,t})\\\
&=\psi_{r}(\pi^{-1}_{1,r}\pi_{2,r})^{-1}\circ
g_{1,r,t}\circ\psi_{t}(\pi^{-1}_{1,t}\pi_{2,t})\\\ &=g_{2,r,t}.\end{array}$

Also,

$\begin{array}[]{ll}g_{2,s(\ast),s}(b_{2,s(\ast)})&=g_{2,s(\ast),s}(\widetilde{\Pi}_{s(\ast)}^{-1}b_{1,s(\ast)})\\\
&=\widetilde{\Pi}_{s}^{-1}g_{1,s(\ast),s}(b_{1,s(\ast)})\\\
&=\widetilde{\Pi}_{s}^{-1}(b_{1}s)\\\ &=b_{2}s.\end{array}$

In sum, we proved that

$x_{2}=(\bar{\pi}_{2},\bar{g}_{2},\bar{b}_{2})\in
X_{{\mathbf{y}}_{{\mathbf{x}}}}({\mathfrak{A}}).$

Next, we are going to show that $x_{2}E_{{\mathfrak{A}}}x_{1}.$ To this end,
let $s\in S$, and set

  * •

$\pi_{s}=\pi_{1,s}^{-1}\pi_{2,s}$,

  * •

$h_{s}=\psi_{s}(\pi_{s}^{-1})$.

Then:

$\begin{array}[]{ll}h_{s}(b_{1,s})&=\psi_{s}(\pi_{2,s}^{-1}\pi_{1,s})(b_{1,s})\\\
&=\widetilde{\Pi}_{s}(g_{1,s(*),s}(b_{1,s(*)}))\\\
&=g_{2,s,s(*)}(\widetilde{\Pi}_{s(*)}b_{1,s(*)})\\\
&=g_{2,s,s(*)}(b_{2,s(*)})\\\ &=b_{2,s},\end{array}$

i.e., $h_{s}(b_{1,s})=b_{2,s}$, from which the equivalence
$x_{2}E_{{\mathfrak{A}}}x_{1}$ follows. ∎

It follows from $(*)_{5.9.1}$ and $(*)_{5.9.2}$ that

${\mathfrak{B}}^{\prime}_{{\mathfrak{A}}}/E_{{\mathfrak{A}}}=Y_{s,\phi}/E_{{\mathfrak{A}}}\simeq{\mathfrak{B}}_{s}.$

Claim 5.9 follows. ∎

Given any ${\mathfrak{A}}\in K_{{\mathbf{x}},1}$, we are going to define an
isomorphism

$F_{{\mathfrak{A}}}:{\mathfrak{A}}\rightarrow{\rm
sort}_{1}({\mathfrak{B}}^{\prime}_{{\mathfrak{A}}}/E_{{\mathfrak{A}}}),$

uniformly definable from ${\mathbf{p}}$. To this end, we set

$F_{{\mathfrak{A}}}(a):=\bigg{\\{}\big{[}(\bar{\pi},\bar{g},\bar{b})\big{]}_{E_{{\mathfrak{A}}}}:(\bar{\pi},\bar{g},\bar{b})\in
X_{{\mathbf{y}}_{{\mathbf{x}}}}({\mathfrak{A}})\text{~{}and~{}}\bigwedge_{s\in
S}\pi_{s}(b_{s})=a\bigg{\\}}.$

In other words,

$F_{{\mathfrak{A}}}(a)=\bigg{\\{}\big{[}(\bar{\pi},\bar{g},\bar{b})\big{]}_{E_{{\mathfrak{A}}}}:(\bar{\pi},\bar{g},\bar{b})\in\mathbb{k}(a)\bigg{\\}}.$

By Claim 5.8, $\mathbb{k}(a)$ is an $E_{{\mathfrak{A}}}$-equivalence class, in
particular, $F_{{\mathfrak{A}}}(a)$ is singleton. Now for ${\mathfrak{A}}\in
K_{{\mathbf{x}},1}$, let ${\mathfrak{B}}_{{\mathfrak{A}}}$ be defined as
follows:

  * •

${\mathfrak{B}}_{{\mathfrak{A}}}$ has as universe $|{\mathfrak{A}}|\cup{\rm
sort}_{2}({\mathfrak{B}}^{\prime}_{{\mathfrak{A}}}/E_{{\mathfrak{A}}})$,

  * •

the mapping $F_{{\mathfrak{A}}}\cup{\rm id}_{{\rm
sort}_{2}({\mathfrak{B}}^{\prime}_{{\mathfrak{A}}}/E_{{\mathfrak{A}}})}$ is an
isomorphism from ${\mathfrak{B}}_{{\mathfrak{A}}}$ onto
${\mathfrak{B}}^{\prime}_{{\mathfrak{A}}}/E_{{\mathfrak{A}}}$.

Then $F({\mathfrak{A}})={\mathfrak{B}}_{{\mathfrak{A}}}$ is the required
function we were looking for, and the theorem follows. ∎

## References

  * [1] J. Adamek, H. Herrlich, J. Rosicky, and W. Tholen, _Injective hulls are not natural_ , Algebra Universalis 48 (2002), no. 4, 379–388.
  * [2] Baumgartner, James E.; _Iterated forcing._ Surveys in set theory, 1–59, London Math. Soc. Lecture Note Ser., 87, Cambridge Univ. Press, Cambridge, 1983.
  * [3] S. Eilenberg and S. Mac Lane, _General theory of natural equivalences_ , Trans. Amer. Math. Soc. 58 (1945) 231-294.
  * [4] H. Fredmam, _On the naturalness of definable operations_ , Houston J. Math. 5 (1979) 325–330.
  * [5] Hodges, Wilfrid; _Model theory_ , Encyclopedia Math. Appl., 42 Cambridge University Press, Cambridge, 1993, xiv+772 pp.
  * [6] Hodges, Wilfrid; _A normal form for algebraic constructions II_ , Logique et Anal. (1975) 429-487; also in Six days of model theory (ed. Paul Henrard, Editions Castella, Albeuve, Switzerland 1977) 197-255.
  * [7] J. Lambek, _Torsion theories, additive semantics, and rings of quotients_ , With an appendix by H. H. Storrer on torsion theories and dominant dimensions. Lecture Notes in Mathematics, 177, Springer-Verlag, Berlin-New York 1971.
  * [8] Hodges, Wilfrid; Shelah, Saharon; _Naturality and definability. I._ J. London Math. Soc. (2) 33 (1986), no. 1, 1–12.
  * [9] Hodges, Wilfrid; Shelah, Saharon; _Naturality and definability II._ Cubo 21 (2019), no. 3, 9–27.
  * [10] Jech, Thomas; _Set theory. The third millennium edition, revised and expanded._ Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003. xiv+769 pp. ISBN: 3-540-44085-2