# Structure of finite dimensional linear bundles morphisms

Fernand PELLETIER 1 and Patrick CABAU 2

###### Abstract

Let $p_{E}:E\to M$ be a fibre bundle over the $m$-dimensional manifold $M$
whose typical fibre is the vector space $\mathbb{R}^{e}$ and let $p_{F}:F\to
N$ be a fibre bundle over the $n$-dimensional manifold $N$ whose typical fibre
is the vector space $\mathbb{R}^{f}$.
We are interested in the structure of the set $\operatorname{Morph}(E,F)$ of
smooth linear bundle morphisms $\Phi:E\to F$ above smooth maps $\varphi:M\to
N$.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries and notations
    1. 2.1 Trivializations and transition functions of a linear bundle
    2. 2.2 Vector bundle homomorphisms
    3. 2.3 Pullback vector bundles
  3. 3 The convenient manifold $C^{\infty}(M,N)$
  4. 4 Structure on $\operatorname{Morph}(E,F)$
    1. 4.1 Associated morphisms to a morphism in $\operatorname{Morph}(E,F)$
    2. 4.2 Structure of convenient manifold on $\operatorname{Morph}(E,F)$

MSC 2010.- 46T05, 46T10.

Keywords.- Convenient manifold, bundle morphisms, pull-back bundles.

Acknowledgments.- The authors would like to thank Pr Eduardo Martínez
(University of Zaragoza, Spain) for having submitted this problem.

1 Univ. Savoie Mont Blanc, CNRS, LAMA, 73000 Chambéry, France
<EMAIL_ADDRESS>
2 Univ. Savoie Mont Blanc, CNRS, LAMA, 73000 Chambéry, France
<EMAIL_ADDRESS>
##  1 Introduction

This note is an answer to a question asked by Pr Eduardo Martínez (University
of Zaragoza, Spain) about the type of structure the set of vector bundle
morphisms between finite dimensional vector bundles over finite dimensional
manifolds could be endowed. In this purpose, according to the book on the
convenient setting (cf. [KrMi97]), given finite dimensional vector bundles
$p_{E}:E\to M$ and $p_{F}:F\to N$, we show that the set
$\operatorname{Morph}(E,F)$ of vector bundle morphisms $\Phi:E\to F$ over
smooth maps $\varphi:M\to N$ can be provided with a structure of convenient
vector bundle $\pi:\operatorname{Morph}(E,F)\to C^{\infty}(M,N)$ over the
convenient manifold of smooth maps $C^{\infty}(M,N)$.

##  2 Preliminaries and notations

In all this note, $M$ (resp. $N$) is a second countable manifold of dimension
$m$ (resp. $n$).

###  2.1 Trivializations and transition functions of a linear bundle

Since $F$ is a vector bunlde, there exists a covering
$\\{V_{\alpha}\\}_{\alpha\in A}$111Where $A$ is a countable set. of $N$ by
open connected sets $V_{\alpha}$ in $N$ such that, for each $\alpha\in A$,
$F_{|V_{\alpha}}=p_{F}^{-1}(V_{\alpha})$ is trivializable. Therefore there
exists a diffeomorphism (trivialization)
$\tau_{\alpha}:p_{F}^{-1}(V_{\alpha})\to V_{\alpha}\times\mathbb{R}^{f}$
satisfying the following conditions:

  1. (TFB1)

$\mathrm{pr}_{1}\circ\tau_{\alpha}=p_{F}$ where
$\operatorname{pr}_{1}:V_{\alpha}\times\mathbb{R}^{f}\to V_{\alpha}$ is the
first projection

$\textstyle{F_{|V_{\alpha}}=p_{F}^{-1}(V_{\alpha})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{\alpha}}$$\scriptstyle{p_{F}}$$\textstyle{V_{\alpha}\times\mathbb{R}^{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{pr}_{1}}$$\textstyle{V_{\alpha}}$

and the restriction $\tau_{\alpha,y}$ of $\tau_{\alpha}$ to the fibre
$F_{y}:=p_{F}^{-1}(y)$ is a linear space isomorphism onto
$\\{y\\}\times\mathbb{R}^{f}$ for all $y\in V_{\alpha}$.

  2. (TFB2)

If $\left(V_{\alpha},\tau_{\alpha}\right)$ and
$\left(V_{\beta},\tau_{\beta}\right)$ are two trivializations such that
$V_{\alpha,\beta}:=V_{\alpha}\cap V_{\beta}\neq\emptyset$, the map

$\tau_{\alpha,\beta,y}=\tau_{\beta,y}\circ\left(\tau_{\alpha,y}\right)^{-1}:\mathbb{R}^{f}\to\mathbb{R}^{f}.$

is an isomorphism for all $y\in V_{\alpha,\beta}$.

  3. (TFB3)

For any trivializations $\left(V_{\alpha},\tau_{\alpha}\right)$ and
$\left(V_{\beta},\tau_{\beta}\right)$ as previously defined, the transition
function $\theta_{\alpha,\beta}$ is characterized by

$\tau_{\beta,y}\circ\left(\tau_{\alpha,y}\right)^{-1}(y,u)=\left(y,\theta_{\alpha,\beta}(y,u)\right).$

and the map $y\mapsto\theta_{\alpha,\beta}(y,.)$ is a smooth map from
$V_{\alpha,\beta}$ to $GL(\mathbb{R}^{f})$.

Moreover, the transition functions fulfil the cocycle conditions:

  1. (CC1)

$\forall y\in V_{\alpha}\cap V_{\beta}\cap
V_{\gamma},\,\theta_{\beta,\gamma}(y)\circ\theta_{\alpha,\beta}(y)=\theta_{\alpha,\gamma}(y)$;

  2. (CC2)

$\forall y\in
V_{\alpha},\,\theta_{\alpha,\alpha}(y)=\operatorname{Id}_{\mathbb{R}^{f}}$.

As previously, there exists a countable covering
$\\{U_{\lambda}\\}_{\lambda\in L}$ of $M$ by open sets $U_{\lambda}$ of $M$
and associated trivializations on the bundle $E$.

###  2.2 Vector bundle homomorphisms

A _vector bundle homomorphism_ $\Phi:E\to F$ which covers $\varphi:M\to N$ is
a smooth map such that

  1. (LBM1)

the following diagram is commutative

$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\scriptstyle{p_{E}}$$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{F}}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{N}$
(2.1)

  2. (LBM2)

$\forall x\in M,\,\Phi_{x}:=\Phi_{|E_{x}}:E_{x}\to F_{\varphi(x)}$ is linear.

As previously, there exists a covering $\\{U_{\lambda},\lambda\in L\\}$ of $M$
by open set $U_{\lambda}$ of $M$ and trivialization
$\sigma_{\lambda}:p_{E}^{-1}(U_{\lambda})\to U_{\lambda}\times\mathbb{R}^{e}$
which satisfies the assumption (TFB1).
Assume that $\varphi(U_{\lambda})\cap V_{\alpha}\not=\emptyset$. If we
restrict the diagram (2.1) to $U_{\lambda}$ and $V_{\alpha}$, the maps
$\Phi_{\lambda}^{\alpha}:=\tau_{\alpha}\circ\Phi\circ\sigma_{\lambda}^{-1}:U_{\lambda}\times\mathbb{R}^{e}\to
V_{\alpha}\times\mathbb{R}^{f}$ can be written

$\Phi_{\lambda}^{\alpha}=\varphi_{|U_{\lambda}}\times A_{\lambda}^{\alpha}$
(2.2)

where $A_{\lambda}^{\alpha}$ is a smooth map from
$U_{\lambda}^{\alpha}:=U_{\lambda}\cap\varphi^{-1}(V_{\alpha})$ to
$GL(\mathbb{R}^{e},\mathbb{R}^{f})$.

###  2.3 Pullback vector bundles

For any smooth function $\varphi:M\to N$, one can define the _pullback vector
bundle_ $p_{F}^{!}:\varphi^{!}F\to M$ of $p_{F}:F\to N$ by

$\varphi^{!}F=\\{(x,v)\in M\times F:\varphi(x)=p_{F}(v)\\},$

whose typical fibre is also $\mathbb{R}^{f}$; we then have a vector bundle
homomorphism $\varphi^{!}$ which covers $\varphi$ defined by

$\varphi^{!}(x,v)=(\varphi(x),v)$ (2.3)

where $v\in F_{\varphi(x)}$ and such that the following diagram is
commutative222cf. [AbTo11]. (LBM!)
$\textstyle{\varphi^{!}F\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi^{!}}$$\scriptstyle{p^{!}_{F}}$$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{F}}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{N}$
and one has $\forall x\in M,\,\varphi^{!}_{x}:\left(\varphi^{!}F\right)_{x}\to
F_{\varphi(x)}$ is an isomorphism (in particular (LBM2) is satisfied).

According to context of $\S$2.1 and $\S$2.2, we have local trivializations
$\sigma_{\lambda}$ of $p_{E}^{-1}(U_{\lambda})$ with $\lambda\in L$, and
$\tau_{\alpha}$ of $p_{F}^{-1}(V_{\alpha})$, with $\alpha\in A$ respectively.
We denote by $U_{\lambda}^{\alpha}=U_{\lambda}\cap\varphi^{-1}(V_{\alpha})$.
Note that $\\{U_{\lambda}^{\alpha}\\}_{(\lambda,\alpha)\in L\times A}$ is a
covering of $M$ such that $E_{|U_{\lambda}^{\alpha}}$ is trivializable with
trivialization $\sigma_{\lambda}$ in restriction to $U_{\lambda}^{\alpha}$ and
$\varphi^{!}F_{|U_{\lambda}^{\alpha}}$ is also trivializable with
trivialization
$(\tau^{!})_{\lambda}^{\alpha}(x,v)=(x,\tau_{\alpha,\varphi(x)}(v))$ in
restriction to $U_{\lambda}^{\alpha}$. Then according to notations given in
$\S$2.2, the expression of $\varphi^{!}$ in the local charts whose domains are
$\left(U_{\lambda}^{\alpha}\times\mathbb{R}^{f}\right)$ and
$\left(V_{\alpha}\times\mathbb{R}^{f}\right)$ is

$(\varphi^{!})_{\lambda}^{\alpha}=\varphi_{|U_{\lambda}^{\alpha}}\times\operatorname{Id}_{\mathbb{R}^{f}}.$
$M$$x$$U_{\lambda}$$U_{\mu}$$U_{\lambda}^{\alpha}$$E$$E_{x}$$u$$p_{E}$$N$$\varphi(x)$$V_{\alpha}$$V_{\beta}$$\varphi$$F$$F_{\varphi(x)}$$v$$p_{F}$$\Phi$$\mathbb{R}^{e}$$u_{\lambda}$$u_{\mu}$$\sigma_{\lambda,x}$$\sigma_{\lambda}$$\mathcal{T}_{\lambda,\mu}$$\mathbb{R}^{f}$$v_{\alpha}$$v_{\beta}$$\tau_{\alpha,\varphi(x)}$$\tau_{\alpha}$$\tau_{\beta}$$\theta_{\alpha,\beta}$$A_{\lambda}^{\alpha}(x)$$A_{\mu}^{\beta}(x)$$\varphi^{!}F$$\left(\varphi^{!}F\right)_{x}$$({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}x},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}v})$$\varphi^{!}$$\Phi^{\varphi}$

$\operatorname{Morph}^{\varphi}$

$p^{!}F$$\left(\tau^{!}\right)_{\lambda}^{\alpha}$

##  3 The convenient manifold $C^{\infty}(M,N)$

Even if the linear space $C^{\infty}\left(\mathtt{U},\mathbb{R}^{n}\right)$,
where $\mathtt{U}$ is an open set of $\mathbb{R}^{m}$, can be endowed with a
structure of Fréchet space (cf. [Bou81], EVT III, 7. Exemples, for $n=1$), it
is not possible to get, in general, such a structure333If the manifold $M$ is
compact, then $C^{\infty}(M,N)$ is metrizable (cf. [KrMi97], 42., Proposition
42.3). on the space $C^{\infty}(M,N)$ of smooth maps from the finite
dimensional manifold $M$ into the finite dimensional manifold $N$.

In fact, following [KrMi97], Theorem 42.1, the space $C^{\infty}(M,N)$ can be
endowed with a structure of convenient manifold444Remark that
$C^{\infty}(M,N)$ can be endowed with other topologies:  $\bullet$ the _strong
$C^{\infty}$ topology_ introduced by Mather (cf. [Ill03]) is important because
various important sets, such as submersions, embeddings,… are open in
$C^{\infty}(M,N)$ (cf. [Com14], Theorem 3.); $\bullet$ the Whitney
$C^{\infty}$ topology $W=\displaystyle\bigcup_{k=0}^{\infty}W_{k}$ where a
basis of neighboroods of the Whitney $C^{k}$ topology is $\\{\varphi\in
C^{\infty}(M,N):\left(j^{k}\varphi\right)(M)\subset U\\}$ where $U$ is an open
set of the $k$-jets bundle $J^{k}(M,N)$. For this topology, $C^{\infty}(M,N)$
is a Baire space. modelled on the space
$C^{\infty}_{c}(M\leftarrow\varphi^{!}TN)$ of compact support sections of the
pullback bundles $\varphi^{!}TN\to M$ along $\varphi:M\to N$ over $M$, endowed
with an adequate bornological topology.

More precisely, since $M$ and $N$ are finite dimensional manifolds, the set of
smooth sections $C^{\infty}(M\leftarrow\varphi^{!}TN)$ has a structure of
convenient space (cf. [KrMi97], 30.1)555According to [KrMi97], Lemma 30.3, if,
moreover, $M$ is a first countable topological space, then
$C^{\infty}(M\leftarrow\varphi^{!}TN)$ has in fact structure of Fréchet
space.. If $K$ is any compact of $M$, the set of smooth sections
$C^{\infty}_{K}(M\leftarrow\varphi^{!}TN)$ with support in $K$ is a closed
subspace of $C^{\infty}(M\leftarrow\varphi^{!}TN)$ and so is a convenient
subspace of $C^{\infty}(M\leftarrow\varphi^{!}TN)$. The space
$C^{\infty}_{c}(M\leftarrow\varphi^{!}TN)$ of smooth sections with compact
support can be endowed with a convenient structure, strict and regular
inductive limit of the spaces $C^{\infty}_{K_{n}}(M\leftarrow\varphi^{!}TN)$
where $\left(K_{n}\right)_{n\in\mathbb{N}}$ is an exhaustive countable base of
compact sets in $M$ (cf. [KrMi97], Lemma 30.4). In particular, a subset in
$C^{\infty}_{c}(M\leftarrow\varphi^{!}TN)$ is $c^{\infty}$ open if and only if
its trace on each such space is open.

Now, let us describe this topology on $C^{\infty}(M,N)$. Consider the
exponential $\exp:\mathbf{U}\to N$ associated to a given Riemannian metric on
$N$, where $\mathbf{U}$ is an open neighbourhood of the zero section in $TN$
such that $\left(\pi_{N},\exp\right):\mathbf{U}\to N\times N$ is a smooth
diffeomorphism onto an open neighbourhood $\mathbf{V}$ of the diagonal.

We then consider the pullback vector bundle

$\textstyle{\varphi^{!}TN\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi^{!}}$$\scriptstyle{\pi_{N}^{!}}$$\textstyle{TN\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{N}}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{N}$

For any $\psi\in C^{\infty}(M,N)$, $\psi\sim\varphi$ means that $\varphi$ and
$\psi$ agree off some compact subset in $M$.
Consider the set

$\mathbf{V}_{\varphi}=\\{\psi\in C^{\infty}(M,N):\forall x\in
M,(\varphi(x),\psi(x))\in\mathbf{V},\psi\sim\varphi\\}$

and the mapping

$\mathbf{v}_{\varphi}:\mathbf{V}_{\varphi}\to
C^{\infty}_{c}(M\leftarrow\varphi^{!}TN)$

defined, for any $\psi\in C^{\infty}(M,N)$ and any $x\in M$, by

$\mathbf{v}_{\varphi}(\psi)(x)=\left(x,\exp^{-1}_{\varphi(x)}(\psi(x))\right).$

######  Remark 3.1.

If $\psi$ belongs to $\mathbf{V}_{\varphi}$, let $\widehat{\psi}\in
C^{\infty}_{c}(M\leftarrow\varphi^{!}TN)$ such that
$(\mathbf{v}_{\varphi}^{-1}(\widehat{\psi}(x))=\psi(x)$. For any $t\in[0,1]$,
consider the time dependent section $\widehat{\psi}_{t}$ defined by
$\widehat{\psi}_{t}(x)=t\widehat{\psi}(x)$, then
$\psi_{t}(x)=\mathbf{v}_{\varphi}^{-1}\left(\widehat{\psi}_{t}(x)\right)$
defines a homotopy from $\varphi$ to $\psi$. Therefore, by homotopy invariance
vector bundles Theorem (cf. for instance [Hus93], Theorem 4.7, p.30),
$\varphi^{!}TN$ and $\psi^{!}TN$ are isomorphic and so are
$C^{\infty}_{c}(M\leftarrow\varphi^{!}TN)$ and
$C^{\infty}_{c}(M\leftarrow\psi^{!}TN)$.

We then endow $C^{\infty}(M,N)$ with a structure of convenient manifold via
the atlas
$\\{\left(\mathbf{V}_{\varphi},\mathbf{v}_{\varphi}\right)\\}_{\varphi\in
C^{\infty}(M,N)}$ where the chart change mappings are given for any
$s\in\mathbf{v}_{\psi}\left(\mathbf{V}_{\varphi}\cap\mathbf{V}_{\psi}\right)$
by

$\displaystyle\left(\mathbf{v}_{\varphi}\circ\mathbf{v}_{\psi}^{-1}\right)(s)=\left(\operatorname{Id}_{M},(\pi_{N},\exp)^{-1}\circ(\varphi,\exp\circ(\pi_{N}^{\ast}\psi)\circ
s)\right).$ (3.1)

##  4 Structure on $\operatorname{Morph}(E,F)$

###  4.1 Associated morphisms to a morphism in $\operatorname{Morph}(E,F)$

In this subsection, the smooth map $\varphi:M\to N$ is fixed.

######  Proposition 4.1.

Let $\Phi:E\to F$ be a vector bundle morphism which covers $\varphi$. The map

$\begin{array}[]{cccc}\Phi^{\varphi}:&E&\to&\varphi^{!}F\\\
&(x,u)&\mapsto&\left(x,(\varphi^{!}_{x})^{-1}\circ\Phi_{x}(u)\right)\end{array}$

is a well defined vector bundle morphism over $\operatorname{Id}_{M}$.
Conversely, if

$\Psi:E\to\varphi^{!}F$

is a vector bundle morphism which covers $\operatorname{Id}_{M}$, the map

$\Phi=\varphi^{!}\circ\Psi$ (4.1)

is a vector bundle morphism which covers $\varphi$. Moreover, we have
$\Phi^{\varphi}=\Psi$.

###### Proof.

First, note that from (2.3), the linear map $(\varphi^{!}_{x})$ is invertible
and so $\Phi^{\varphi}(x,u)$ is well defined. We must show that
$\Phi^{\varphi}$ is a smooth map. Since it is a local problem, we restrict
ourselves to the context of $\S$2.2 and $\S$2.3; choose any $x\in M$ such that
$x$ belongs to $U_{\lambda}$ and $\varphi(x)$ belongs to some $V_{\alpha}$; we
then have $U_{\lambda}^{\alpha}\not=\emptyset$. Then the bundles
$E_{|U_{\lambda}^{\alpha}}$, $F_{|V_{\alpha}}$ and
$\varphi^{!}F_{|U_{\lambda}^{\alpha}}$ are trivializable. Then, from the
definition of $\Phi^{\varphi}$, and according to (2.2), we have:

$(\Phi^{\varphi})_{\lambda}^{\alpha}:=\tau_{\alpha}\circ\Phi^{\varphi}\circ\left((\tau^{!})_{\lambda}^{\alpha}\right)^{-1}=\left(\operatorname{Id}_{U_{\lambda}^{\alpha}},\tau_{\alpha,\varphi(x)}\circ\Phi\circ(\sigma_{\lambda,x}^{-1})_{|U_{\lambda}^{\alpha}\times\mathbb{R}^{e}}\right)$
(4.2)

with $\sigma_{\lambda,x}=\operatorname{pr}_{2}^{e}\circ\sigma_{\lambda}$ and
$\tau_{\alpha,\varphi(x)}=\operatorname{pr}_{2}^{f}\circ\tau_{\alpha}$ where
$\operatorname{pr}_{2}^{e}$ (resp. $\operatorname{pr}_{2}^{f}$) is the
projection on the typical type fibre $\mathbb{R}^{e}$ (resp.
$\mathbb{R}^{f}$). So we get
$\left(\Phi^{\varphi}\right)_{\lambda}^{\alpha}=\left(\operatorname{Id}_{U_{\lambda}^{\alpha}},A_{\lambda}^{\alpha}\right)$
and then $(\Phi^{\varphi})_{\lambda}^{\alpha}$ is a smooth map, for any such
open sets $U_{\lambda}^{\alpha}$ and $V_{\alpha}$. This implies that
$\Phi^{\varphi}$ is a smooth map. Finally, from its definition,
$\Phi^{\varphi}$ satisfies the condition (LBM!).

For the converse, the relation (4.1) defines a vector bundle morphism as a
composition of such morphisms and it is clear that we have
$\Phi^{\varphi}=\Psi$. ∎

From Proposition 4.1, we have a bijection

$\begin{array}[]{cccc}\operatorname{Morph}^{\varphi}:&\operatorname{Morph}_{\varphi}(E,F)&\to&\operatorname{Morph}_{\operatorname{Id}_{M}}\left(E,\varphi^{!}F\right)\\\
&\Phi&\mapsto&\Phi^{\varphi}\end{array}$ (4.3)

where

  1. $\bullet$

$\operatorname{Morph}_{\operatorname{Id}_{M}}\left(E,\varphi^{!}F\right)$ is
the set of vector bundle morphisms: $E\to\varphi^{!}F$ above
$\operatorname{Id}_{M}$

  2. $\bullet$

$\operatorname{Morph}_{\varphi}(E,F)$ is the set of vector bundle morphisms:
$E\to F$ above $\varphi$.

It is easy to see that $L(E,\varphi^{!}F):=\displaystyle\bigcup_{x\in
M}L\left(E_{x},F_{\varphi(x)}\right)$ has a structure of vector bundle over
$M$ whose typical fibre is $L\left(\mathbb{R}^{e},\mathbb{R}^{f}\right)$ and
structural group $GL\left(\mathbb{R}^{e}\right)\times
GL\left(\mathbb{R}^{f}\right)$. So
$\operatorname{Morph}_{\operatorname{Id}_{M}}\left(E,\varphi^{!}F\right)$ is
the space of smooth sections of $L(E,\varphi^{!}F)\to M$ and, according to
[KrMi97], 30, this set is denoted $C^{\infty}\left(M\leftarrow
L\left(E,\varphi^{!}F\right))\right)$ and has a structure of Fréchet space
from Proposition 30.1.

######  Remark 4.2.

  1. 1.

Note that $\operatorname{Morph}_{\varphi}(E,F)$ and
$\operatorname{Morph}_{\operatorname{Id}_{M}}\left(E,\varphi^{!}F\right)$ have
a real vector space structure and it is easy to see that
$\operatorname{Morph}^{\varphi}$ is linear. Since
$\operatorname{Morph}^{\varphi}$ is a bijection, then we can provide
$\operatorname{Morph}_{\varphi}(E,F)$ with a structure of convenient space and
so $\operatorname{Morph}^{\varphi}$ is an isomorphism.

  2. 2.

Assume that $\psi$ is homotopic to $\varphi$. Then by Homotopy invariance
vector bundles Theorem (cf. [Hus93], Theorem 4.7, p.30), there exists a bundle
isomorphism $H_{\varphi,\psi}$ from $\varphi^{!}F$ to $\psi^{!}F$, and so we
get an isomorphism from
$\operatorname{Morph}_{\operatorname{Id}_{M}}\left(E,\varphi^{!}F\right)$ to
$\operatorname{Morph}_{\operatorname{Id}_{M}}\left(E,\psi^{!}F\right)$ given
by
$\operatorname{Morph}^{\psi}=H_{\varphi,\psi}\circ\operatorname{Morph}^{\varphi}$.

###  4.2 Structure of convenient manifold on $\operatorname{Morph}(E,F)$

Consider

$\begin{array}[]{cccc}\pi:&\operatorname{Morph}(E,F)&\to&C^{\infty}(M,N)\\\
&(\varphi,\Phi)&\mapsto&\varphi\end{array}$

where one has

$\operatorname{Morph}(E,F)=\displaystyle\bigcup_{\varphi\in
C^{\infty}(M,N)}\operatorname{Morph}_{\operatorname{Id}_{M}}\left(E,\varphi^{!}F\right)$

We will provide a convenient manifold structure on $\operatorname{Morph}(E,F)$
such that $\pi:\operatorname{Morph}(E,F)\to C^{\infty}(M,N)$ is a convenient
vector bundle.

######  Theorem 4.3.

There exists a convenient manifold structure on $\operatorname{Morph}(E,F)$
modelled on $C^{\infty}_{c}(M\leftarrow\varphi^{!}TN)\times
C^{\infty}\left(M\leftarrow L\left(E,\varphi^{!}F\right)\right)$.
Moreover,

$\pi:\operatorname{Morph}(E,F)\to C^{\infty}(M,N)$

is a convenient vector bundle with typical fibre
$\operatorname{Morph}_{\operatorname{Id}_{M}}\left(E,\varphi^{!}F\right)$.

######  Remark 4.4.

According to [KrMi97], 42.4, Theorem 4.3 is also true if $N$ has a structure
of convenient manifold which can be provided with a local addition. On the
other hand, if $M$ is compact, then $C^{\infty}(M,N)$ can be endowed with a
structure of Fréchet manifold and $\operatorname{Morph}_{\varphi}(E,F)$ is a
Fréchet space; so $\pi:\operatorname{Morph}(E,F)\to C^{\infty}(M,N)$ is a
Fréchet vector bundle.

###### Proof.

We consider a chart $\left(\mathbf{V}_{\varphi},\mathbf{v}_{\varphi}\right)$
around $\varphi$ in $C^{\infty}(M,N)$ and according to Remark 4.2, consider
the map

$\begin{array}[]{cccc}\mathbf{T}^{\varphi}:\par&\pi^{-1}(\mathbf{V}_{\varphi})\par&\to\par&\mathbf{V}_{\varphi}\times\operatorname{Morph}_{\operatorname{Id}_{M}}\par\left(E,\varphi^{!}F\right)\\\
&(\psi,\Psi)\par&\mapsto\par&\big{(}\psi,\left(H_{\varphi,\psi}\right)^{-1}\circ\operatorname{Morph}^{\psi}(\Psi)\big{)}\par\end{array}$

We then get the following commutative diagram:

$\textstyle{\operatorname{Morph}(E,F)_{\mathbf{V}_{\varphi}}=\pi^{-1}(\mathbf{V}_{\varphi})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbf{T}^{\varphi}}$$\scriptstyle{\pi}$$\textstyle{\mathbf{V}_{\varphi}\times\operatorname{Morph}_{\operatorname{Id}_{M}}\left(E,\varphi^{!}F\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{pr}_{1}}$$\textstyle{\mathbf{V}_{\varphi}}$

From the definition of $\mathbf{T}^{\varphi}$, its restriction to
$\pi^{-1}(\psi)=\operatorname{Morph}_{\psi}(E,F)$ is
$\left(H_{\varphi,\psi}\right)^{-1}\circ\operatorname{Morph}^{\psi}(\Psi)$.
According to Remark 4.2, $H_{\varphi,\psi}$ is a diffeomorphism from
$\varphi^{!}F$ to $\psi^{!}F$ and so
$H_{\varphi,\psi}^{-1}\circ\operatorname{Morph}^{\psi}(\Psi)$ is a
diffeomorphism from $\operatorname{Morph}_{\psi}(E,F)$ to
$\operatorname{Morph}_{\operatorname{Id}_{M}}\left(E,\varphi^{!}F\right)$.
It follows that the map $\mathbf{T}^{\varphi}$ is bijective and so we can
provide $\operatorname{Morph}(E,F)_{\mathbf{V}_{\varphi}}$ with the convenient
manifold structure modelled on $C^{\infty}_{c}(M\leftarrow\varphi^{!}TN)\times
C^{\infty}\left(M\leftarrow L(\mathbb{R}^{e},\mathbb{R}^{f})\right)$ for which
$\mathbf{T}^{\varphi}$ is a diffeomorphism. Note that, since the restriction
$\mathbf{T}^{\varphi}$ to $\pi^{-1}(\psi)$ is
$\left(H_{\varphi,\psi}\right)^{-1}\circ\operatorname{Morph}^{\psi}(\Psi)$,
the convenient structure of manifold on each $\pi^{-1}(\psi)$ defined in
Remark 4.2, Assertion 1 is a supplemented convenient submanifold of
$\operatorname{Morph}(E,F)_{\mathbf{V}_{\varphi}}$. This implies that
$\mathbf{T}^{\varphi}$ is a trivialization of $\pi^{-1}(\mathbf{V}_{\varphi})$
and the restriction of $\pi$ to this convenient manifold is a surjective
submersion.
Consider a chart $\mathbf{V}_{\psi}$ in $C^{\infty}(M,N)$ such that
$\mathbf{V}_{\psi}\cap\mathbf{V}_{\varphi}\not=\emptyset$.

The transition functions

$\mathbf{T}^{\varphi}\circ(\mathbf{T}^{\psi})^{-1}:\;\mathbf{T}^{\psi}\Big{(}\left(\mathbf{V}_{\varphi}\cap\mathbf{V}_{\psi}\right)\times\operatorname{Morph}_{\operatorname{Id}_{M}}\left(E,\psi^{!}F\right)\Big{)}\to\mathbf{T}^{\varphi}\Big{(}\left(\mathbf{V}_{\varphi}\cap\mathbf{V}_{\psi}\right)\times\operatorname{Morph}_{\operatorname{Id}_{M}}\left(E,\varphi^{!}F\right)\Big{)}$

are the maps given by

$\mathbf{T}^{\varphi}\circ(\mathbf{T}^{\psi})^{-1}=\left(\mathbf{v}_{\varphi}\circ\mathbf{v}_{\psi}^{-1},(\operatorname{Morph}^{\varphi})(\Phi)^{-1}\circ(\operatorname{Morph}^{\psi}(\Psi))\right).$
(4.4)

where $\mathbf{v}_{\varphi}\circ\mathbf{v}_{\psi}^{-1}$ corresponds to the
change of charts (3.1).
As we have already seen,

$\Pi_{\varphi}:\operatorname{Morph}_{\operatorname{Id}_{M}}\left(E,\varphi^{!}F\right)\to
M:=C^{\infty}\left(M\leftarrow L(\mathbb{R}^{e},\mathbb{R}^{f})\right)$

has a convenient vector space structure. Then if
$\left\\{\left(\mathbf{V}_{\varphi},\mathbf{v}\right)\right\\}_{\varphi\in
C^{\infty}(M,N)}$ is an atlas on $C^{\infty}(M,N)$, it follows that
$\left\\{\left(\pi^{-1}((\mathbf{V}_{\varphi})),(\mathbf{v}_{\varphi},\mathbf{T}_{\phi})\right)\right\\}$
is an atlas which defines a convenient structure manifold on
$\operatorname{Morph}(E,F)$ modelled on
$C^{\infty}_{c}(M\leftarrow\varphi^{!}TN)\times\operatorname{Morph}_{\operatorname{Id}_{M}}\left(E,\varphi^{!}F\right)$
and $\pi:\operatorname{Morph}(E,F)\to C^{\infty}(M,N)$ is also a convenient
vector bundle with typical fibre
$\operatorname{Morph}_{\operatorname{Id}_{M}}\left(E,\varphi^{!}F\right)$. ∎

######  Remark 4.5.

Since $E$ and $F$ are finite dimensional fibre bundles over $M$ and $N$
respectively, $E$ and $F$ are in particular second countable finite
dimensional manifolds, the set $C^{\infty}(E,F)$ can be provided with a
convenient structure of manifold modelled on
$C^{\infty}_{c}(E\leftarrow\varphi^{\ast}TF)$, as proved in $\S$3 for the set
$C^{\infty}(M,N)$ of smooth maps between second countable finite-dimensional
manifolds.

As $\operatorname{Morph}(E,F)$ is a subset of $C^{\infty}(E,F)$ which is
provided with its own structure of convenient manifold, it is natural to ask
us if, in this context,

_Is $\operatorname{Morph}(E,F)$ a weak convenient submanifold of
$C^{\infty}(E,F)$?_

Recall that if $X_{1}$ and $X_{2}$ are convenient manifolds respectively
modelled on the convenient spaces $\mathbb{X}_{1}$ and $\mathbb{X}_{2}$, a
smooth map $f:X_{1}\to X_{2}$ is called a weak immersion if, for any $x\in
X_{1}$, there exist a chart $(U_{1},\phi_{1})$ around $x\in X_{1}$ and a chart
$(U_{2},\phi_{2})$ around $f(x)\in X_{2}$ such that

$\phi_{2}\circ f\circ\phi_{1}^{-1}=\iota_{\phi_{1}(U_{1})}$ (4.5)

where $\iota:\mathbb{X}_{1}\to\mathbb{X}_{2}$ is a smooth linear injective
map.

Then $\iota$ is continuous for the convenient topology, i.e., for any
$c^{\infty}$ open set $\mathbb{U}_{2}\subset\mathbb{X}_{2}$,
$\iota^{-1}(\mathbb{U}_{2})$ is a $c^{\infty}$-open set in $\mathbb{X}_{1}$.
This implies that the inclusion of $X_{1}$ into $X_{2}$ is a smooth injective
and continuous map between $X_{1}$ and $X_{2}$. Now, if, moreover, $Z$ is a
weak submanifold of $X_{1}$, since the composition of smooth convenient maps
is a smooth map and the composition of continuous maps is continuous, the
inclusion of $Z$ into $X_{2}$ is also an injective convenient smooth
continuous map from $Z$ to $X_{2}$. We will apply the argument for the
inclusion of the fibre $\pi^{-1}(\varphi)$ to $C^{\infty}(E,F)$.

_For an answer to the above question, argue by contradiction: assume that the
answer is yes._

Let $\Phi$ be a bundle morphism from $E$ to $F$, there exists a chart
$(\mathbf{W}_{\Phi},\mathbf{w}_{\Phi})$ around $\Phi$, and, for any
$\Psi\in\mathbf{W}_{\Phi}$, there exists a compact subset $K_{\Psi}$ in $E$
such that $\Phi=\Psi$ on $E\setminus K_{\Psi}$. In particular, if $\Phi$
(resp. $\Psi$) covers $\varphi$ (resp. $\psi$) then $\varphi=\psi$ on
$p_{E}(E\setminus K_{\Psi})\subset M$. Now we have seen in the proof of
Theorem 4.2 that $\pi^{-1}(\varphi)$ is a submanifold of
$\operatorname{Morph}(E,F)$. So, accoding to the previous general argument,
under our assumption,
$\operatorname{Morph}_{\varphi}(E,F)\cap\mathbf{W}_{\Phi}$, for some chart
domain $\mathbf{W}_{\Phi}$ in $C^{\infty}(E,F)$, is open in
$\operatorname{Morph}_{\varphi}(E,F)$. Therefore, there exist vector bundle
morphisms $\Phi$ and $\Psi$ which cover $\varphi$, such that $\Phi\not=\Psi$
and which belongs to
$\operatorname{Morph}_{\varphi}(E,F)\cap\mathbf{W}_{\Phi}$. But, there exists
a compact set $K_{\Psi}$ in $E$ such that $\Phi=\Psi$ on $E\setminus
K_{\Psi}$. But since $\Psi\not=\Phi$, there must exist $(x,u)\in K_{\Psi}$
such that $\Phi(x,u)\not=\Psi(x,u)$. However, there exists
$n\in\mathbb{N}\setminus\\{0\\}$ such that $(x,nu)\not\in K_{\Psi}$ and so
$\Psi(x,nu)=\Phi(x,nu)$. Since $\Psi$ is linear in the second variable, one
has $n\Psi(x,u)=n\Phi(x,u)$, so we get $\Psi(x,u)=\Phi(x,u)$ because $n\neq
0$. Now, this result is also true for any $x\in\pi(K_{\Psi})$ and this implies
that $\Psi=\Phi$, which means that
$\operatorname{Morph}_{\varphi}(E,F)\cap\mathbf{W}_{\Phi}=\\{\Phi\\}$ which is
not an open set of $\operatorname{Morph}_{\varphi}(E,F)$. By the above general
argument, we get a contradiction. Therefore, $\operatorname{Morph}(E,F)$ is
not a weak convenient submanifold of $C^{\infty}(E,F)$.

## References

  * [AbTo11] M. Abate, F. Tovena, Geometria differenziale. Unitext, 54, Springer, Milan, 2011.
  * [Bou81] N. Bourbaki, Espaces vectoriels topologiques. Chapitres 1 à 5, Éléments de Mathématique, Masson, Paris, 1981.
  * [Cer61] J. Cerf, Topologie de certains espaces de plongements. Bull. Soc. Math. France 89 (1961) 227–380.
  * [Com14] E. Compaan, Function Space Topologies. Preprint University of Illinois at Urbana-Champain, 2014.
  * [Hus93] D. Husemoller Fibre Bundles. Graduate Texts in Mathematics 20 springer, Third Edition (1993).
  * [Ill03] S. Illmann, The very-strong topology on $C^{\infty}(M,N)$ and $K$-equivariant maps. Osaka J. Math. 40 (2003) 409–428.
  * [KrMi97] A. Kriegl, P.W. Michor, The convenient setting of global analysis. Mathematical Surveys and Monographs, 53. American Mathematical Society, Providence, RI, 1997.
  * [Mat69] J. Mather, Stability of $C^{\infty}$-mappings: II. Infinitesimal stability implies stability. Ann. of Math. 89 (1969) 254–291.
  * [Stee51] N.E. Steenrod, The topology of fibre bundles. Princeton Mathematical Series 14, Princeton Univ. Press, 1951.
  * [Vil03] C. Villani, Analyse II. Cours de deuxième année, E.N.S. Lyon (2003).
  * [Whi34] H. Whitney, Analytic extensions of differentiable functions defined in closed sets. Trans. Amer. Math. Soc. 36 (1934) 63–89.


s^{\prime}_{i}(\overline{x},y)\\}&\models_{\mathsf{V_{\rhd}}}t^{\prime}_{i}(\overline{x},y)\vee
v^{\prime}(\overline{x},\overline{z})\\\ &\iff\forall k\in
K_{i}\colon\\{u^{\prime}(\overline{x},\overline{z})\wedge
s^{\prime\prime}_{i,k}(\overline{x})\\}\models_{\mathsf{V_{\rhd}}}t^{\prime\prime\prime}_{i,k}(\overline{x})\vee
v^{\prime}(\overline{x},\overline{z}).\end{split}$ (16)

The left-to-right direction of this equivalence is immediate. For the converse
direction,

$\displaystyle\forall k\in
K_{i}\colon\\{u^{\prime}(\overline{x},\overline{z})\wedge
s^{\prime\prime}_{i,k}(\overline{x})\\}\models_{\mathsf{V_{\rhd}}}t^{\prime\prime\prime}_{i,k}(\overline{x})\vee
v^{\prime}(\overline{x},\overline{z})$ $\displaystyle\Longrightarrow\forall
k\in K_{i}\colon\\{u^{\prime}(\overline{x},\overline{z})\wedge
s^{\prime\prime}_{i,k}(\overline{x})\\}\models_{\mathsf{V}}t^{\prime\prime\prime}_{i,k}(\overline{x})\vee
v^{\prime}(\overline{x},\overline{z})$ Prop. 7.1
$\displaystyle\Longrightarrow\\{u^{\prime}(\overline{x},\overline{z})\wedge
s^{\prime}_{i}(\overline{x},y)\\}\models_{\mathsf{V}}t^{\prime}_{i}(\overline{x},y)\vee
w(\overline{x})\vee v^{\prime}(\overline{x},\overline{z})$ by (15)
$\displaystyle\Longrightarrow\\{u^{\prime}(\overline{x},\overline{z})\wedge
s^{\prime}_{i}(\overline{x},y)\\}\models_{\mathsf{V}}t^{\prime}_{i}(\overline{x},y)\vee
v^{\prime}(\overline{x},\overline{z})$ since $\\{w(\overline{x})\wedge
s^{\prime}_{i}(\overline{x},y)\\}\models_{\mathsf{V}}t^{\prime}_{i}(\overline{x},y)$
$\displaystyle\Longrightarrow\\{u^{\prime}(\overline{x},\overline{z})\wedge
s^{\prime}_{i}(\overline{x},y)\\}\models_{\mathsf{V_{\rhd}}}t^{\prime}_{i}(\overline{x},y)\vee
v^{\prime}(\overline{x},\overline{z})$
$\displaystyle\text{Prop.~{}\ref{p:vvt}}.$

We define now

$t^{\star}(\overline{x})\coloneqq\bigwedge_{i\in I}\bigwedge_{k\in
K_{i}}\left(s^{\prime\prime}_{i,k}(\overline{x})\rhd
t^{\prime\prime\prime}_{i,k}(\overline{x})\right).$

Consider any
$u(\overline{x},\overline{z})\in\mathrm{Tm}_{\mathcal{L}_{\rhd}}(\overline{x},\overline{z})$.
By Lemma 7.5 (with $t=\Lambda$), there exist a finite non-empty set $J$ and
$u^{\prime}_{j}(\overline{x},\overline{z})\in\mathrm{Tm}_{\mathcal{L}}(\overline{x},\overline{z})$,
$v^{\prime}_{j}(\overline{x},\overline{z})\in\mathrm{Tm}_{\mathcal{L}}(\overline{x},\overline{z})\cup\\{\Lambda\\}$
($j\in J$) such that for any
$s^{\prime}(\overline{x},y),t^{\prime}(\overline{x},y)\in\mathrm{Tm}_{\mathcal{L}_{\rhd}}(\overline{x},y)$,

$\\{u(\overline{x},\overline{z})\wedge
s^{\prime}(\overline{x},y)\\}\models_{\mathsf{V_{\rhd}}}t^{\prime}(\overline{x},y)\iff\forall
j\in J\colon\\{u^{\prime}_{j}(\overline{x},\overline{z})\wedge
s^{\prime}(\overline{x},y)\\}\models_{\mathsf{V_{\rhd}}}t^{\prime}(\overline{x},y)\vee
v^{\prime}_{j}(\overline{x},\overline{z}).$

In particular, for each $i\in I$ and $k\in K_{i}$,

$\displaystyle\begin{split}\\{u(\overline{x},\overline{z})\wedge
s^{\prime}_{i}(\overline{x},y)\\}&\models_{\mathsf{V_{\rhd}}}t^{\prime}_{i}(\overline{x},y)\\\
&\iff\forall j\in J\colon\\{u^{\prime}_{j}(\overline{x},\overline{z})\wedge
s^{\prime}_{i}(\overline{x},y)\\}\models_{\mathsf{V_{\rhd}}}t^{\prime}_{i}(\overline{x},y)\vee
v^{\prime}_{j}(\overline{x},\overline{z})\end{split}$ (17)
$\displaystyle\begin{split}\\{u(\overline{x},\overline{z})\wedge
s^{\prime\prime}_{i,k}(\overline{x})\\}&\models_{\mathsf{V_{\rhd}}}t^{\prime\prime\prime}_{i,k}(\overline{x})\\\
&\iff\forall j\in J\colon\\{u^{\prime}_{j}(\overline{x},\overline{z})\wedge
s^{\prime\prime}_{i,k}(\overline{x})\\}\models_{\mathsf{V_{\rhd}}}t^{\prime\prime\prime}_{i,k}(\overline{x})\vee
v^{\prime}_{j}(\overline{x},\overline{z}).\end{split}$ (18)

Putting these equivalences together, it follows that

$\displaystyle\\{u(\overline{x},\overline{z})\\}$
$\displaystyle\models_{\mathsf{V_{\rhd}}}t(\overline{x},y)$
$\displaystyle\iff\forall i\in I\colon\\{u(\overline{x},\overline{z})\wedge
s^{\prime}_{i}(\overline{x},y)\\}\models_{\mathsf{V_{\rhd}}}t^{\prime}_{i}(\overline{x},y)$
by (14) $\displaystyle\iff\forall i\in I,j\in
J\colon\\{u^{\prime}_{j}(\overline{x},\overline{z})\wedge
s^{\prime}_{i}(\overline{x},y)\\}\models_{\mathsf{V_{\rhd}}}t^{\prime}_{i}(\overline{x},y)\vee
v^{\prime}_{j}(\overline{x},\overline{z})$ by (17) $\displaystyle\iff\forall
i\in I,j\in J,k\in
K_{i}\colon\\{u^{\prime}_{j}(\overline{x},\overline{z})\wedge
s^{\prime\prime}_{i,k}(\overline{x})\\}\models_{\mathsf{V_{\rhd}}}t^{\prime\prime\prime}_{i,k}(\overline{x})\vee
v^{\prime}_{j}(\overline{x},\overline{z})$ by (16) $\displaystyle\iff\forall
i\in I,k\in K_{i}\colon\\{u(\overline{x},\overline{z})\wedge
s^{\prime\prime}_{i,k}(\overline{x})\\}\models_{\mathsf{V_{\rhd}}}t^{\prime\prime\prime}_{i,k}(\overline{x})$
by (18) $\displaystyle\iff\forall i\in I,k\in
K_{i}\colon\\{u(\overline{x},\overline{z})\\}\models_{\mathsf{V_{\rhd}}}s^{\prime\prime}_{i,k}(\overline{x})\rhd
t^{\prime\prime\prime}_{i,k}(\overline{x})$ Prop. 7.2
$\displaystyle\iff\\{u(\overline{x},\overline{z})\\}\models_{\mathsf{V_{\rhd}}}t^{\star}(\overline{x}).$
$\displaystyle\qed\proofend$

∎

In particular, the conditions of Theorem 7.6 are satisfied when $\mathsf{V}$
is the variety $\mathsf{LA}$ of lattice-ordered abelian groups.

######  Theorem 7.7.

The theory of $\mathsf{LA_{\rhd}}$ has a model completion.

###### Proof.

Recall that $\mathsf{LA}$ is generated as a quasivariety by the lattice-
ordered abelian group ${\bf{R}}=\langle\mathbb{R},\min,\max,+,-,0\rangle$
(see, e.g., [AF88]). Hence, to show that $\mathsf{LA_{\rhd}}$ has a model
completion, it suffices to check the condition of Theorem 7.6 with
$\models_{\mathsf{V}}$ replaced by $\models_{{\bf{R}}}$.

We proceed in two stages. Suppose first that $\overline{x},y$ is a finite set,
$s(\overline{x},y)\in\mathrm{Tm}_{\mathcal{L}}(\overline{x},y)$ is a meet of
group terms, and
$t(\overline{x},y)\in\mathrm{Tm}_{\mathcal{L}}(\overline{x},y)\cup\\{\Lambda\\}$
is $\Lambda$ or a join of group terms. We prove that there exist
$s^{\prime}(\overline{x})\in\mathrm{Tm}_{\mathcal{L}}(\overline{x})$,
$t^{\prime}(\overline{x})\in\mathrm{Tm}_{\mathcal{L}}(\overline{x})\cup\\{\Lambda\\}$
satisfying for any
$u(\overline{x},\overline{z}),v(\overline{x},\overline{z})\in\mathrm{Tm}_{\mathcal{L}}(\overline{x},\overline{z})$,

$\\{u(\overline{x},\overline{z})\wedge
s(\overline{x},y)\\}\models_{{\bf{R}}}t(\overline{x},y)\vee
v(\overline{x},\overline{z})\iff\\{u(\overline{x},\overline{z})\wedge
s^{\prime}(\overline{x})\\}\models_{{\bf{R}}}t^{\prime}(\overline{x})\vee
v(\overline{x},\overline{z}).$

It is easily checked in ${\bf{R}}$ that for any $\mathcal{L}$-terms $u,v,w$
and $k\in\mathbb{N}^{>0}$,

$\\{u\wedge v\\}\models_{{\bf{R}}}w\iff\\{u\wedge
kv\\}\models_{{\bf{R}}}w\enspace\text{and}\enspace\\{u\\}\models_{{\bf{R}}}v\vee
w\iff\\{u\\}\models_{{\bf{R}}}v\vee kw.$

Also, ${\bf{R}}\models x+(y\wedge z)\approx(x+y)\wedge(x+z)$ and
${\bf{R}}\models x+(y\vee z)\approx(x+y)\vee(x+z)$. Hence, reasoning in
${\bf{R}}$, we may assume that there exist a $k\in\mathbb{N}^{>0}$ and
$s_{0}(\overline{x}),s_{1}(\overline{x}),s_{2}(\overline{x}),t_{1}(\overline{x}),t_{2}(\overline{x})\in\mathrm{Tm}_{\mathcal{L}}(\overline{x})$,
$t_{0}(\overline{x})\in\mathrm{Tm}_{\mathcal{L}}(\overline{x})\cup\\{\Lambda\\}$
such that $s(\overline{x},y)$ is the meet of $s_{0}(\overline{x})$ and some
members (possibly none) of
$\\{s_{1}(\overline{x})+ky,s_{2}(\overline{x})-ky\\}$, and $t(\overline{x},y)$
is the join of $t_{0}(\overline{x})$ and some members (possibly none) of
$\\{t_{1}(\overline{x})+ky,t_{2}(\overline{x})-ky\\}$. We construct
$s^{\prime}(\overline{x})\in\mathrm{Tm}_{\mathcal{L}}(\overline{x})$,
$t^{\prime}(\overline{x})\in\mathrm{Tm}_{\mathcal{L}}(\overline{x})\cup\\{\Lambda\\}$
as follows. Let $s^{\prime}(\overline{x})\coloneqq
s_{0}(\overline{x})\wedge(s_{1}(\overline{x})+s_{2}(\overline{x}))$ if
$s_{1}(\overline{x})+ky$ and $s_{2}(\overline{x})-ky$ occur in
$s(\overline{x},y)$; otherwise $s^{\prime}(\overline{x})\coloneqq
s_{0}(\overline{x})$. The term $t^{\prime}(\overline{x})$ is the join of
$t_{0}(\overline{x})$ and (i) $t_{1}(\overline{x})+t_{2}(\overline{x})$ if
$t_{1}(\overline{x})+ky$ and $t_{2}(\overline{x})-ky$ occur in
$t(\overline{x},y)$; (ii) $t_{1}(\overline{x})-s_{1}(\overline{x})$ if
$s_{1}(\overline{x})+ky$ occurs in $s(\overline{x},y)$ and
$t_{1}(\overline{x})+ky$ occurs in $t(\overline{x},y)$; (iii)
$t_{2}(\overline{x})-s_{2}(\overline{x})$ if $s_{2}(\overline{x})-ky$ occurs
in $s(\overline{x},y)$ and $t_{2}(\overline{x})-ky$ occurs in
$t(\overline{x},y)$.

We check the case where
$s(\overline{x},y)=(s_{1}(\overline{x})+ky)\wedge(s_{2}(\overline{x})-ky)$,
$t(\overline{x},y)=(t_{1}(\overline{x})+ky)\vee(t_{2}(\overline{x})-ky)$,
$s^{\prime}(\overline{x})=s_{1}(\overline{x})+s_{2}(\overline{x})$, and
$t^{\prime}(\overline{x})=(t_{1}(\overline{x})+t_{2}(\overline{x}))\vee(t_{1}(\overline{x})-s_{1}(\overline{x}))\vee(t_{2}(\overline{x})-s_{2}(\overline{x}))$,
other cases being very similar. Let
$u(\overline{x},\overline{z}),v(\overline{x},\overline{z})\in\mathrm{Tm}_{\mathcal{L}}(\overline{x},\overline{z})$
and suppose first that $\\{u(\overline{x},\overline{z})\wedge
s^{\prime}(\overline{x})\\}\models_{{\bf{R}}}t^{\prime}(\overline{x})\vee
v(\overline{x},\overline{z})$. Since
$\\{s(\overline{x},y)\\}\models_{{\bf{R}}}s^{\prime}(\overline{x})$ and
$\\{s(\overline{x})\wedge
t^{\prime}(\overline{x})\\}\models_{{\bf{R}}}t(\overline{x},y)$, it follows
that $\\{u(\overline{x},\overline{z})\wedge
s(\overline{x},y)\\}\models_{{\bf{R}}}t(\overline{x},y)\vee
v(\overline{x},\overline{z})$. Now suppose that
$\\{u(\overline{x},\overline{z})\wedge
s^{\prime}(\overline{x})\\}\not\models_{{\bf{R}}}t^{\prime}(\overline{x})\vee
v(\overline{x},\overline{z})$; that is, some assignment
$f\colon\overline{x},\overline{z}\to\mathbb{R}$ satisfies
$0\leq\tilde{f}(u(\overline{x},\overline{z}))$,
$0\leq\tilde{f}(s_{1}(\overline{x}))+\tilde{f}(s_{2}(\overline{x}))$,
$\tilde{f}(t_{1}(\overline{x}))+\tilde{f}(t_{2}(\overline{x}))<0$,
$\tilde{f}(t_{1}(\overline{x}))<\tilde{f}(s_{1}(\overline{x}))$,
$\tilde{f}(t_{2}(\overline{x}))<\tilde{f}(s_{2}(\overline{x}))$, and
$\tilde{f}(v(\overline{x},\overline{z}))<0$. We extend $f$ to an assignment
$g\colon\overline{x},y,\overline{z}\to\mathbb{R}$ by defining

$g(y)\coloneqq\frac{\min{(\tilde{f}(s_{2}),-\tilde{f}(t_{1}))}+\max{(-\tilde{f}(s_{1}),\tilde{f}(t_{2}))}}{2k}.$

We obtain $0\leq\tilde{g}(u(\overline{x},\overline{z}))$,
$0\leq\tilde{g}(s_{1}(\overline{x}))+\tilde{g}(ky)$,
$\tilde{g}(ky)\leq\tilde{g}(s_{2}(\overline{x}))$,
$\tilde{g}(t_{1}(\overline{x}))+\tilde{g}(ky)<0$,
$\tilde{g}(t_{2}(\overline{x}))<\tilde{g}(ky)$, and
$\tilde{g}(v(\overline{x},\overline{z}))<0$. So
$\\{u(\overline{x},\overline{z})\wedge
s(\overline{x},y)\\}\not\models_{{\bf{R}}}t(\overline{x},y)\vee
v(\overline{x},\overline{z})$.

We now prove that for any finite set $\overline{x},y$ and
$s(\overline{x},y)\in\mathrm{Tm}_{\mathcal{L}}(\overline{x},y)$,
$t(\overline{x},y)\in\mathrm{Tm}_{\mathcal{L}}(\overline{x},y)\cup\\{\Lambda\\}$,
there exist a finite non-empty set $K$ and
$s^{\prime}_{k}(\overline{x})\in\mathrm{Tm}_{\mathcal{L}}(\overline{x})$,
$t^{\prime}_{k}(\overline{x})\in\mathrm{Tm}_{\mathcal{L}}(\overline{x})\cup\\{\Lambda\\}$
($k\in K$) satisfying for any
$u(\overline{x},\overline{z}),v(\overline{x},\overline{z})\in\mathrm{Tm}_{\mathcal{L}}(\overline{x},\overline{z})$,

$\\{u(\overline{x},\overline{z})\wedge
s(\overline{x},y)\\}\models_{{\bf{R}}}t(\overline{x},y)\vee
v(\overline{x},\overline{z})\iff\forall k\in
K\colon\\{u(\overline{x},\overline{z})\wedge
s^{\prime}_{k}(\overline{x})\\}\models_{{\bf{R}}}t^{\prime}_{k}(\overline{x})\vee
v(\overline{x},\overline{z}).$

By the distributivity properties of ${\bf{R}}$, we may assume that
$s(\overline{x},y)$ is $s_{1}(\overline{x},y)\vee\cdots\vee
s_{n}(\overline{x},y)$, where each $s_{i}(\overline{x},y)$ is a meet of group
terms, and that $t(\overline{x},y)$ is
$t_{1}(\overline{x},y)\wedge\cdots\wedge t_{m}(\overline{x},y)$, where either
each $t_{j}(\overline{x},y)$ is a join of group terms or $m=1$ and
$t_{1}(\overline{x},y)=\Lambda$. Using the first part of this proof, for each
$i\in\\{1,\dots,n\\}$ and $j\in\\{1,\dots,m\\}$, there exist
$s^{\prime}_{i,j}(\overline{x})\in\mathrm{Tm}_{\mathcal{L}}(\overline{x})$ and
$t^{\prime}_{i,j}(\overline{x})\in\mathrm{Tm}_{\mathcal{L}}(\overline{x})\cup\\{\Lambda\\}$
satisfying for any
$u(\overline{x},\overline{z}),v(\overline{x},\overline{z})\in\mathrm{Tm}_{\mathcal{L}}(\overline{x},\overline{z})$,

$\\{u(\overline{x},\overline{z})\wedge
s_{i}(\overline{x},y)\\}\models_{{\bf{R}}}t_{j}(\overline{x},y)\vee
v(\overline{x},\overline{z})\iff\\{u(\overline{x},\overline{z})\wedge
s^{\prime}_{i,j}(\overline{x})\\}\models_{{\bf{R}}}t^{\prime}_{i,j}(\overline{x})\vee
v(\overline{x},\overline{z}).$

Hence for any
$u(\overline{x},\overline{z}),v(\overline{x},\overline{z})\in\mathrm{Tm}_{\mathcal{L}}(\overline{x},\overline{z})$,

$\displaystyle\\{u(\overline{x},\overline{z})\wedge
s(\overline{x},y)\\}\models_{{\bf{R}}}t(\overline{x},y)\vee
v(\overline{x},\overline{z})\iff$ $\displaystyle\forall
i\in\\{1,\dots,n\\},j\in\\{1,\dots,m\\}\colon$
$\displaystyle\\{u(\overline{x},\overline{z})\wedge
s^{\prime}_{i,j}(\overline{x})\\}\models_{{\bf{R}}}t^{\prime}_{i,j}(\overline{x})\vee
v(\overline{x},\overline{z}).$ $\displaystyle\qed\proofend$

∎

The variety $\mathsf{MV_{\rhd}}$ of MV-algebras extended with $\rhd$ is term-
equivalent to the variety $\mathsf{MV}_{\\!\Delta}$ of MVΔ-algebras (see,
e.g., [Haj98, MP01]), generated by linearly ordered MV-algebras extended with
an additional unary operator $\Delta$ defined by

$\Delta x\coloneqq\begin{cases}1&\text{if }x=1\\\
0&\text{otherwise.}\end{cases}$

For any algebra in $\mathsf{MV_{\rhd}}$, we let $\Delta x\coloneqq(x\rhd
0)\rhd 0$, and for any algebra in $\mathsf{MV}_{\\!\Delta}$, we let $x\rhd
y\coloneqq\lnot\Delta x\oplus y$. The following theorem may therefore be
proved along the same lines as Theorem 7.7, using the fact that $\mathsf{MV}$
is generated as a quasivariety by the algebra
$\langle[0,1],\oplus,\lnot,0\rangle$, where $x\oplus y\coloneqq\min{(1,x+y)}$
and $\lnot x\coloneqq 1-x$ (see, e.g., [COM99]).

######  Theorem 7.8.

The theory of $\mathsf{MV}_{\\!\Delta}$ has a model completion.

A different proof of this last result (obtained by giving an explicit
axiomatization) was presented by X. Caicedo at the conference Residuated
Structures: Algebra and Logic in 2008. Caicedo also presented a proof (again
by giving an explicit axiomatization) at the Latin American Algebra Colloquium
in 2019 that the theory of the class of so-called “pseudo-complemented”
lattice-ordered abelian groups has a model completion, which seems to bear
some relation to our Theorem 7.7.

### Acknowledgement

The authors would like to thank the anonymous referee for their careful
reading of the paper and valuable comments.

##  Appendix A A comparison with Wheeler’s characterization theorem

In this appendix, we relate our results in Section 3 to the necessary and
sufficient conditions provided by Wheeler in [Whe76] (see also [Whe78]) for
the existence of model completions for universal theories with finite
presentations. Let us note that Wheeler considers arbitrary first-order
languages and, although we restrict ourselves here to algebraic languages, it
is not difficult to see that the results of this appendix can be generalized
to any first-order language by replacing equations with atomic formulas.

Let $\mathsf{K}$ be a universal class of algebras. Following [Whe76, Section
3], a _finite presentation_ of an algebra ${\bf{A}}\in\mathsf{K}$ is a pair
$(\overline{a},\pi)$ where $\overline{a}\in A^{\overline{x}}$ is a finite set
of generators for ${\bf{A}}$ and $\pi(\overline{x})$ is a conjunction of
equations such that ${\bf{A}}\models\pi(\overline{a})$ and for any equation
$\varepsilon(\overline{x})$,

${\bf{A}}\models\varepsilon(\overline{a})\iff\mathsf{K}\models\pi\to\varepsilon.$

Observe that $(\overline{a},\pi)$ is a finite presentation of ${\bf{A}}$ if,
and only if, for any ${\bf{B}}\in\mathsf{K}$ generated by $\overline{b}\in
B^{\overline{x}}$ satisfying ${\bf{B}}\models\pi(\overline{b})$, there is a
surjective homomorphism ${\bf{A}}\twoheadrightarrow{\bf{B}}$ mapping
$\overline{a}$ to $\overline{b}$. An algebra ${\bf{A}}\in\mathsf{K}$ is said
to be _finitely presented_ in $\mathsf{K}$ if it admits a finite presentation.

######  Remark A.1.

If $\mathsf{K}$ is a variety, then an algebra ${\bf{A}}\in\mathsf{K}$ is
finitely presented in $\mathsf{K}$ if, and only if, it is finitely presented
in the usual sense, i.e., ${\bf{A}}$ is isomorphic to the quotient of a
finitely generated $\mathsf{K}$-free algebra with respect to a compact
congruence. Moreover, if $\mathsf{K}$ is a positive universal class and the
variety $\mathsf{V}$ generated by $\mathsf{K}$ is congruence distributive,
then ${\bf{A}}\in\mathsf{K}$ is finitely presented in $\mathsf{K}$ if, and
only if, it is finitely presented in $\mathsf{V}$. Just observe that by
Jónsson’s Lemma [Jon67], we have in this case
$\mathsf{V}=\mathbb{ISP}(\mathsf{K})$.

A class of algebras $\mathsf{K}$ has _finite presentations_ if for every
finite set $\overline{x}$ and conjunction of equations $\pi(\overline{x})$
that is satisfiable in $\mathsf{K}$, there exist an algebra
${\bf{A}}\in\mathsf{K}$ and a tuple $\overline{a}\in A^{\overline{x}}$ such
that $(\overline{a},\pi)$ is a presentation of ${\bf{A}}$. All quasivarieties,
and in particular all varieties, have finite presentations (cf. [Whe76,
Corollary 1, p. 315]), but this is not the case for all universal classes, as
shown by the following example.

######  Example A.2.

The positive universal class $\mathsf{LA^{c}}$ of ordered abelian groups does
not have finite presentations. To see this, recall that $\mathsf{LA^{c}}$
generates the variety $\mathsf{LA}$ of lattice-ordered abelian groups, which
is congruence distributive (cf. Example 2.8). Hence, by Remark A.1, an ordered
abelian group is finitely presented in the sense of the above definition if,
and only if, it is finitely presented as a lattice-ordered abelian group in
the usual sense. However, any finitely presented ordered abelian group is
simple (see, e.g., [Glass1999, Theorem 4.A and Corollary 5.2.3]) and there
exist finitely generated ordered abelian groups that are not simple, e.g., the
lexicographic product ${\bf{R}}\overrightarrow{\times}{\bf{R}}$ generated by
$\\{(1,0),(0,1)\\}$. It follows that there cannot be a $2$-generated finitely
presented ordered abelian group with presentation $\top(x_{1},x_{2})$.

Let us now recall Wheeler’s conservative congruence extension property. Given
any algebra ${\bf{A}}\in\mathsf{K}$, let $\mathfrak{D}^{+}({\bf{A}})$ be the
positive diagram of ${\bf{A}}$, i.e., the set of atomic sentences in the
language extended with names for the elements of ${\bf{A}}$ that are satisfied
in ${\bf{A}}$. The class $\mathsf{K}$ has the _conservative congruence
extension property (for finite presentations)_ if, whenever ${\bf{B}}$ admits
a finite presentation
$((\overline{a},\overline{b}),\pi(\overline{x},\overline{y}))$, ${\bf{A}}$ is
the subalgebra of ${\bf{B}}$ generated by $\overline{a}$, the tuple
$\overline{b}$ does not lie in ${\bf{A}}$, and
$\rho(\overline{x},\overline{y})$ is a conjunction of negated equations such
that ${\bf{B}}\models\rho(\overline{a},\overline{b})$, there exists a
quantifier-free formula $\chi(\overline{x})$ satisfying

$\mathrm{Th}(\mathsf{K})\cup\mathfrak{D}^{+}({\bf{B}})\vdash\rho(\overline{a},\overline{b})\to\chi(\overline{a}),$

and for every surjective homomorphism
$h\colon{\bf{A}}\twoheadrightarrow{\bf{A}}^{\prime}$ such that
${\bf{A}}^{\prime}\in\mathsf{K}$ and
${\bf{A}}^{\prime}\models\chi(h(\overline{a}))$ there exist a
${\bf{B}}^{\prime}\in\mathsf{K}$ extending ${\bf{A}}^{\prime}$ and a
surjective homomorphism
$h^{\prime}\colon{\bf{B}}\twoheadrightarrow{\bf{B}}^{\prime}$ whose
restriction to ${\bf{A}}$ coincides with $h$, and such that
${\bf{B}}^{\prime}\models\rho(h(\overline{a}),h^{\prime}(\overline{b}))$.444Note
that Wheeler does not assume that ${\bf{A}}^{\prime}\in\mathsf{K}$, but uses
this in the proof of his main result.

The following proposition shows that, for universal classes with finite
presentations, the conservative congruence extension property is equivalent to
a strengthening of the conservative model extension property where the
variable $y$ is replaced by a tuple $\overline{y}$.

######  Proposition A.3.

Let $\mathsf{K}$ be a universal class of algebras with finite presentations.
Then $\mathsf{K}$ has the conservative congruence extension property if, and
only if, for any finite sets $\overline{x},\overline{y}$ and conjunction of
literals $\psi(\overline{x},\overline{y})$, there exists a quantifier-free
formula $\chi(\overline{x})$ satisfying

  1. (i)

$\mathsf{K}\models\psi\to\chi$

  2. (ii)

for every ${\bf{A}}\in\mathsf{K}$ generated by $\overline{a}\in
A^{\overline{x}}$ such that ${\bf{A}}\models\chi(\overline{a})$ and for any
equation $\varepsilon(\overline{x})$,

$\mathsf{K}\models\psi^{+}\to\varepsilon\>\Longrightarrow\>{\bf{A}}\models\varepsilon(\overline{a}),$

there exist an algebra ${\bf{B}}\in\mathsf{K}$ extending ${\bf{A}}$ and
$\overline{b}\in B^{\overline{y}}$ such that
${\bf{B}}\models\psi(\overline{a},\overline{b})$.

###### Proof.

Suppose that $\mathsf{K}$ has the conservative congruence extension property.
Fix finite sets $\overline{x},\overline{y}$ and a conjunction of literals
$\psi(\overline{x},\overline{y})$. If $\psi$ is not satisfiable in
$\mathsf{K}$, we can set $\chi\coloneqq\bot$. Hence, assume that $\psi$ is
satisfiable in $\mathsf{K}$. Consider ${\bf{B}}\in\mathsf{K}$ and
$(\overline{a},\overline{b})\in B^{\overline{x},\overline{y}}$ such that
$((\overline{a},\overline{b}),\psi^{+})$ is a finite presentation of
${\bf{B}}$ and let ${\bf{A}}$ be the subalgebra of ${\bf{B}}$ generated by
$\overline{a}$. If there is a $b_{i}\in\overline{b}$ such that $b_{i}\in A$,
then $\mathsf{K}\models\psi^{+}\to y_{i}\approx t(\overline{x})$ for some term
$t$. Replacing $y_{i}$ by $t(\overline{x})$ in the formula $\psi$ whenever
$b_{i}\in A$, we can assume that no element of $\overline{b}$ belongs to
${\bf{A}}$. Further, if ${\bf{B}}\models\sigma(\overline{a},\overline{b})$ for
some equation $\sigma$ of $\psi^{-}$, then
$\mathsf{K}\models\psi^{+}\to\sigma$, contradicting the fact that $\psi$ is
satisfiable in $\mathsf{K}$. Hence
${\bf{B}}\models\rho(\overline{a},\overline{b})$, where
$\rho(\overline{x},\overline{y})$ is the conjunction of the negated equations
$\neg\sigma$ for $\sigma$ ranging over the equations of $\psi^{-}$ (hence,
$\psi=\psi^{+}\and\rho$), and so there exists a quantifier-free formula
$\chi(\overline{x})$ satisfying the conditions for the conservative congruence
extension property.

We prove that $\chi$ satisfies (i) and (ii). For (i), consider any algebra
${\bf{C}}\in\mathsf{K}$ and tuples $\overline{c}\in C^{\overline{x}}$ and
$\overline{d}\in C^{\overline{y}}$ such that
${\bf{C}}\models\psi(\overline{c},\overline{d})$. We must prove that
${\bf{C}}\models\chi(\overline{c})$. Let ${\bf{C}}^{\prime}$ be the subalgebra
of ${\bf{C}}$ generated by $\overline{c},\overline{d}$, and note that
${\bf{C}}^{\prime}\models\psi(\overline{c},\overline{d})$ as $\psi$ is
quantifier-free. Since
${\bf{C}}^{\prime}\models\psi^{+}(\overline{c},\overline{d})$, there is a
surjective homomorphism ${\bf{B}}\twoheadrightarrow{\bf{C}}^{\prime}$ mapping
$\overline{a}$ to $\overline{c}$ and $\overline{b}$ to $\overline{d}$. Hence,
with the obvious interpretation of the new constants,
${\bf{C}}^{\prime}\models\mathfrak{D}^{+}({\bf{B}})$. So
$\mathrm{Th}(\mathsf{K})\cup\mathfrak{D}^{+}({\bf{B}})\vdash\rho\to\chi$ and
${\bf{C}}^{\prime}\models\rho(\overline{c},\overline{d})$ entail
${\bf{C}}^{\prime}\models\chi(\overline{c})$, and therefore
${\bf{C}}\models\chi(\overline{c})$.

For (ii), consider any algebra ${\bf{A}}^{\prime}\in\mathsf{K}$ generated by a
tuple $\overline{a}^{\prime}\in(A^{\prime})^{\overline{x}}$ such that
${\bf{A}}^{\prime}\models\chi(\overline{a}^{\prime})$ and, for all equations
$\varepsilon(\overline{x})$, $\mathsf{K}\models\psi^{+}\to\varepsilon$ entails
${\bf{A}}^{\prime}\models\varepsilon(\overline{a}^{\prime})$. Then there
exists a (unique) homomorphism $h\colon{\bf{A}}\to{\bf{A}}^{\prime}$ mapping
$\overline{a}$ to $\overline{a}^{\prime}$. Just observe that, whenever
$s(\overline{a})=t(\overline{a})$ for two terms
$s(\overline{x}),t(\overline{x})$, we have ${\bf{B}}\models
s(\overline{a})\approx t(\overline{a})$ and hence also
$\mathsf{K}\models\psi^{+}\to s(\overline{x})\approx t(\overline{x})$, which
implies ${\bf{A}}^{\prime}\models s(\overline{a}^{\prime})\approx
t(\overline{a}^{\prime})$. Moreover, $h$ is surjective because
$\overline{a}^{\prime}$ generates ${\bf{A}}^{\prime}$. By the conservative
congruence extension property, there exist ${\bf{B}}^{\prime}\in\mathsf{K}$
extending ${\bf{A}}^{\prime}$ and a surjective homomorphism
$h^{\prime}\colon{\bf{B}}\twoheadrightarrow{\bf{B}}^{\prime}$ that extends $h$
and satisfies
${\bf{B}}^{\prime}\models\rho(\overline{a}^{\prime},h^{\prime}(\overline{b}))$.
Using the fact that ${\bf{B}}\models\psi^{+}(\overline{a},\overline{b})$
entails
${\bf{B}}^{\prime}\models\psi^{+}(\overline{a}^{\prime},h^{\prime}(\overline{b}))$,
we conclude that
${\bf{B}}^{\prime}\models\psi(\overline{a}^{\prime},h^{\prime}(\overline{b}))$,
as was to be proved.

For the converse direction, let ${\bf{B}}\in\mathsf{K}$ be an algebra
admitting a finite presentation
$((\overline{a},\overline{b}),\pi(\overline{x},\overline{y}))$, let ${\bf{A}}$
be the subalgebra of ${\bf{B}}$ generated by $\overline{a}$, and assume that
$\overline{b}$ does not lie in ${\bf{A}}$. Further, let
$\rho(\overline{x},\overline{y})$ be a conjunction of negated equations such
that ${\bf{B}}\models\rho(\overline{a},\overline{b})$. Define the conjunction
of literals $\psi(\overline{x},\overline{y})\coloneqq\pi\and\rho$. Then there
exists a quantifier-free formula $\chi(\overline{x})$ satisfying the
properties in (i) and (ii). Condition (i) entails easily that

$\mathrm{Th}(\mathsf{K})\cup\mathfrak{D}^{+}({\bf{B}})\vdash\rho(\overline{a},\overline{b})\to\chi(\overline{a}).$

Consider next a surjective homomorphism
$h\colon{\bf{A}}\twoheadrightarrow{\bf{A}}^{\prime}$ with
${\bf{A}}^{\prime}\in\mathsf{K}$ and
${\bf{A}}^{\prime}\models\chi(h(\overline{a}))$. Note that $h(\overline{a})$
generates ${\bf{A}}^{\prime}$ and, for any equation
$\varepsilon(\overline{x})$, $\mathsf{K}\models\pi\to\varepsilon$ implies
${\bf{B}}\models\varepsilon(\overline{a})$ and hence also
${\bf{A}}\models\varepsilon(\overline{a})$ and
${\bf{A}}^{\prime}\models\varepsilon(h(\overline{a}))$. By (ii), there exist
${\bf{B}}^{\prime}\in\mathsf{K}$ extending ${\bf{A}}^{\prime}$ and a tuple
$\overline{b}^{\prime}\in(B^{\prime})^{\overline{y}}$ such that
${\bf{B}}^{\prime}\models\psi(h(\overline{a}),\overline{b}^{\prime})$. Let
${\bf{B}}^{\prime\prime}$ be the subalgebra of ${\bf{B}}^{\prime}$ generated
by $h(\overline{a}),\overline{b}^{\prime}$ and note that
${\bf{B}}^{\prime\prime}\in\mathsf{K}$ because $\mathsf{K}$ is a universal
class. Since
${\bf{B}}^{\prime\prime}\models\pi(h(\overline{a}),\overline{b}^{\prime})$,
there exists a surjective homomorphism
$h^{\prime}\colon{\bf{B}}\twoheadrightarrow{\bf{B}}^{\prime\prime}$ mapping
$\overline{a}$ to $h(\overline{a})$ and $\overline{b}$ to
$\overline{b}^{\prime}$. In particular, $h^{\prime}$ extends $h$. Further,
${\bf{B}}^{\prime\prime}$ extends ${\bf{A}}^{\prime}$ and satisfies
${\bf{B}}^{\prime\prime}\models\rho(h(\overline{a}),\overline{b}^{\prime})$.
Hence $\mathsf{K}$ has the conservative congruence extension property. ∎

We are now in a position to compare Wheeler’s characterization of universal
theories that have a model completion with our results from Section 3.
Following Wheeler, we say that a universal class of algebras with finite
presentations $\mathsf{K}$ is _coherent_ if, whenever ${\bf{B}}\in\mathsf{K}$
is finitely presented in $\mathsf{K}$ and ${\bf{A}}$ is a finitely generated
subalgebra of ${\bf{B}}$, then ${\bf{A}}$ is finitely presented in
$\mathsf{K}$. It is not difficult to see that a universal class of algebras
with finite presentations is coherent if, and only if, it has the variable
projection property.

In the case where $\mathsf{K}$ is a universal class of algebras, the following
main result of [Whe76] is a direct consequence of Theorem 3.2 and Propositions
3.8 and A.3.

######  Proposition A.4 ([Whe76, Theorem 5]).

Let $\mathsf{K}$ be a universal class with finite presentations. The theory of
$\mathsf{K}$ has a model completion if, and only if, $\mathsf{K}$ is coherent
and has the amalgamation property and conservative congruence extension
property.

## References

  * [1] AF88 M.M.E.Anderson and T.T.H.Feil 19880 M.E. Anderson T.H. Feil Lattice-ordered groups: An introduction, Springer,1988. anderson me feil th 1988 lattice ordered groups an introduction
  * [2] Bak74 K.K.A.Baker 19740 K.A. Baker Primitive satisfaction and equational problems for lattices and other algebras, Trans. Amer. Math. Soc., vol.190(1974), pp.125–150. baker ka 1974 primitive satisfaction and equational problems for lattices and other algebras
  * [3] BKP84 W.W.J.Blok, P.P.Köhler, and D.D.Pigozzi 19840 W.J. Blok, P. Köhler, D. Pigozzi On the structure of varieties with equationally definable principal congruences II, Algebra Universalis, vol.18(1984), pp.334–379. blok wj kohler p pigozzi d 1984 on the structure of varieties with equationally definable principal congruences ii
  * [4] BP86 W.W.J.Blok and D.D.Pigozzi 19860 W.J. Blok D. Pigozzi A finite basis theorem for quasivarieties, Algebra Universalis, vol.22(1986), no.1, pp.1–13. blok wj pigozzi d 1986 finite basis theorem for quasivarieties
  * [5] BT03 K.K.Blount and C.C.Tsinakis 20030 K. Blount C. Tsinakis The structure of residuated lattices, Int. J. Algebr. Comput., vol.13(2003), no.4, pp.437–461. blount k tsinakis c 2003 structure of residuated lattices
  * [6] BKLT16 M.M.Botur, J.J.Kühr, L.L.Liu, and C.C.Tsinakis 20160 M. Botur, J. Kühr, L. Liu, C. Tsinakis The Conrad program: from $\ell$-groups to algebras of logic, J. Algebra, vol.450(2016), pp.173–203. botur m kuhr j liu l tsinakis c 2016 conrad program from ell groups to algebras of logic
  * [7] BS81 S.S.Burris and H.H.P.Sankappanavar 19810 S. Burris H.P. Sankappanavar A course in universal algebra, Springer,1981. burris s sankappanavar hp 1981 course in universal algebra
  * [8] GhilardiEtAl2021 D.D.Calvanese, S.S.Ghilardi, A.A.Gianola, M.M.Montali, and A.A.Rivkin 20210 D. Calvanese, S. Ghilardi, A. Gianola, M. Montali, A. Rivkin Model completeness, uniform interpolants and superposition calculus (with applications to verification of data-aware processes), J. Automat. Reason., vol.65(2021), no.7, pp.941–969. calvanese d ghilardi s gianola a montali m rivkin a 2021 model completeness uniform interpolants and superposition calculus with applications to verification of data aware processes
  * [9] CK90 C.C.C.Chang and H.H.J.Keisler 19900 C.C. Chang H.J. Keisler Model theory, third ed., Studies in Logic and the Foundations of Mathematics, vol. 73, North-Holland Publishing Co., Amsterdam,1990. chang cc keisler hj 1990 model theory
  * [10] COM99 R.R.Cignoli, I. M. L.I. M. L.D’Ottaviano, and D.D.Mundici 19990 R. Cignoli, I. M. L. D’Ottaviano, D. Mundici Algebraic foundations of many-valued reasoning, Kluwer,1999. cignoli r dottaviano i m l mundici d 1999 algebraic foundations of many valued reasoning
  * [11] DJ2018 L.L.Darnière and M.M.Junker 20180 L. Darnière M. Junker Model completion of varieties of co-Heyting algebras, Houston J. Math., vol.44(2018), no.1, pp.49–82. darniere l junker m 2018 model completion of varieties of co heyting algebras
  * [12] Day1971 A.A.Day 1971/720 A. Day A note on the congruence extension property, Algebra Universalis, vol.1(1971/72), pp.234–235. day a 197172 note on the congruence extension property
  * [13] ES70 P.P.Eklof and G.G.Sabbagh 19700 P. Eklof G. Sabbagh Model-completions and modules, Ann. Math. Logic, vol.2(1970), no.3, pp.251–295. eklof p sabbagh g 1970 model completions and modules
  * [14] FGQ80 E.E.Fried, G.G.Grätzer, and R.R.Quackenbush 19800 E. Fried, G. Grätzer, R. Quackenbush Uniform congruence schemes, Algebra Universalis, vol.10(1980), no.2, pp.176–188. fried e gratzer g quackenbush r 1980 uniform congruence schemes
  * [15] Gal03 N.N.Galatos 20030 N. Galatos Varieties of residuated lattices, Ph.D. thesis, Vanderbilt University, Nashville,2003. galatos n 2003 varieties of residuated lattices
  * [16] GJKO07 N.N.Galatos, P.P.Jipsen, T.T.Kowalski, and H.H.Ono 20070 N. Galatos, P. Jipsen, T. Kowalski, H. Ono Residuated lattices: An algebraic glimpse at substructural logics, Elsevier,2007. galatos n jipsen p kowalski t ono h 2007 residuated lattices an algebraic glimpse at substructural logics
  * [17] Geh14 M.M.Gehrke 20140 M. Gehrke Canonical extensions, Esakia spaces, and universal models, Leo Esakia on duality in modal and intuitionistic logics, Outst. Contrib. Log., vol. 4, Springer,2014, pp.9–41. gehrke m 2014 canonical extensions esakia spaces and universal models
  * [18] GH01 M.M.Gehrke and J.J.Harding 20010 M. Gehrke J. Harding Bounded lattice expansions, J. Algebra, vol.238(2001), no.1, pp.345–371. gehrke m harding j 2001 bounded lattice expansions
  * [19] GZ97 S.S.Ghilardi and M.M.Zawadowski 19970 S. Ghilardi M. Zawadowski Model completions and r-Heyting categories, Ann. Pure Appl. Logic, vol.88(1997), no.1, pp.27–46. ghilardi s zawadowski m 1997 model completions and r heyting categories
  * [20] GZ02 S.S.Ghilardi and M.M.Zawadowski 20020 Sheaves, games, and model completions: A categorical approach to nonclassical propositional logics, Springer,2002. ghilardi s zawadowski m 2002 sheaves games and model completions a categorical approach to nonclassical propositional logics
  * [21] Glass1999 A.A.M.W.Glass 19990 A.M.W. Glass Partially ordered groups, Series in Algebra, vol. 7, World Scientific Publishing Co., Inc., River Edge, NJ,1999. glass amw 1999 partially ordered groups
  * [22] GP80 A.A.M.W.Glass and K.K.R.Pierce 19800 A.M.W. Glass K.R. Pierce Existentially complete abelian lattice-ordered groups, Trans. Amer. Math. Soc., vol.261(1980), no.1, pp.255–270. glass amw pierce kr 1980 existentially complete abelian lattice ordered groups
  * [23] Haj98 P.P.Hájek 19980 P. Hájek Metamathematics of fuzzy logic, Kluwer,1998. hajek p 1998 metamathematics of fuzzy logic
  * [24] Hod97 W.W.Hodges 19970 W. Hodges A shorter model theory, Cambridge University Press, Cambridge,1997. hodges w 1997 shorter model theory
  * [25] Jon67 B.B.Jónsson 19670 B. Jónsson Algebras whose congruence lattices are distributive, Math. Scand., vol.21(1967), pp.110–121 (1968). jonsson b 1967 algebras whose congruence lattices are distributive
  * [26] KP1980 P.P.Köhler and D.D.Pigozzi 19800 P. Köhler D. Pigozzi Varieties with equationally definable principal congruences, Algebra Universalis, vol.11(1980), no.2, pp.213–219. kohler p pigozzi d 1980 varieties with equationally definable principal congruences
  * [27] KM18 T.T.Kowalski and G.G.Metcalfe 20190 T. Kowalski G. Metcalfe Uniform interpolation and coherence, Ann. Pure Appl. Logic, vol.170(2019), no.7, pp.825–841. kowalski t metcalfe g 2019 uniform interpolation and coherence
  * [28] Lac79 F.F.Lacava 19790 F. Lacava Alcune proprietà delle Ł-algebre e delle Ł-algebre esistenzialmente chiuse, Boll. Un. Mat. Ital. A (5), vol.16(1979), no.2, pp.360–366. lacava f 1979 alcune proprieta delle l algebre e delle l algebre esistenzialmente chiuse
  * [29] LS77 F.F.Lacava and D.D.Saeli 19770 F. Lacava D. Saeli Sul model-completamento della teoria delle Ł-catene, Boll. Un. Mat. Ital. A (5), vol.14(1977), no.1, pp.107–110. lacava f saeli d 1977 sul model completamento della teoria delle l catene
  * [30] MMT14 G.G.Metcalfe, F.F.Montagna, and C.C.Tsinakis 20140 G. Metcalfe, F. Montagna, C. Tsinakis Amalgamation and interpolation in ordered algebras, J. Algebra, vol.402(2014), pp.21–82. metcalfe g montagna f tsinakis c 2014 amalgamation and interpolation in ordered algebras
  * [31] MPT10 G.G.Metcalfe, F.F.Paoli, and C.C.Tsinakis 20100 G. Metcalfe, F. Paoli, C. Tsinakis Ordered algebras and logic, Uncertainty and rationality (H. Hosni F. Montagna, editors), Publications of the Scuola Normale Superiore di Pisa, Vol. 10,2010, pp.1–85. metcalfe g paoli f tsinakis c 2010 ordered algebras and logic
  * [32] Millar1995 T.T.Millar 19950 T. Millar Model completions and omitting types, J. Symbolic Logic, vol.60(1995), no.2, pp.654–672. millar t 1995 model completions and omitting types
  * [33] MP01 F.F.Montagna and G.G.Panti 20010 F. Montagna G. Panti Adding structure to MV-algebras, J. Pure Appl. Algebra, vol.164(2001), pp.365–387. montagna f panti g 2001 adding structure to mv algebras
  * [34] Pie72 K.K.R.Pierce 19720 K.R. Pierce Amalgamations of lattice-ordered groups, Trans. Amer. Math. Soc., vol.172(1972), pp.249–260. pierce kr 1972 amalgamations of lattice ordered groups
  * [35] Pit92 A.A.M.Pitts 19920 A.M. Pitts On an interpretation of second-order quantification in first-order intuitionistic propositional logic, J. Symbolic Logic, vol.57(1992), pp.33–52. pitts am 1992 on an interpretation of second order quantification in first order intuitionistic propositional logic
  * [36] Rob77 A.A.Robinson 19770 A. Robinson Complete theories, second ed., North-Holland Publishing Co., Amsterdam-New York-Oxford,1977, With a preface by H.J. Keisler, Studies in Logic and the Foundations of Mathematics. robinson a 1977 complete theories
  * [37] TZ2012 K.K.Tent and M.M.Ziegler 20120 K. Tent M. Ziegler A course in model theory, Lecture Notes in Logic, Cambridge University Press,2012. tent k ziegler m 2012 course in model theory
  * [38] Whe76 W.W.H.Wheeler 19760 W.H. Wheeler Model-companions and definability in existentially complete structures, Israel J. Math., vol.25(1976), no.3, pp.305–330. wheeler wh 1976 model companions and definability in existentially complete structures
  * [39] Whe78 W.W.H.Wheeler 19780 A characterization of companionable, universal theories, J. Symbolic Logic, vol.43(1978), no.3, pp.402–429. wheeler wh 1978 characterization of companionable universal theories
  * [40]