# Automorphisms and derivations on algebras endowed with formal infinite sums

Vincent Bagayoko
IMJ-PRG  Lothar Sebastian Krapp
Universität Konstanz  Salma Kuhlmann
Universität Konstanz  Daniel Panazzolo
Université de Haute Alsace  Michele Serra
TU Dortmund

###### Abstract

We establish a correspondence between automorphisms and derivations on certain
algebras of generalised power series. In particular, we describe a Lie algebra
of derivations on a field $k(\\!(G)\\!)$ of generalised power series,
exploiting our knowledge of its group of valuation preserving automorphisms.
The correspondence is given by the formal Taylor expansion of the exponential.
In order to define the exponential map, we develop an appropriate notion of
summability of infinite families in algebras. We show that there is a large
class of algebras in which the exponential induces the above correspondence.

## Introduction

Let $k$ be a field of characteristic $0$. The automorphism group of the valued
field $k(\\!(t)\\!)$ of Laurent series was studied in Schilling’s classical
paper [23]. A derivation on $k(\\!(t)\\!)$ is a $k$-linear map $\partial\colon
k(\\!(t)\\!)\longrightarrow k(\\!(t)\\!)$ satisfying the Leibniz product rule.
The $k$-vector space of derivations on $k(\\!(t)\\!)$ becomes a Lie algebra
once endowed with the Lie bracket
$[\partial_{1},\partial_{2}]:=\partial_{1}\circ\partial_{2}-\partial_{2}\circ\partial_{1}$.
Let $\sigma$ and $\partial$ denote respectively an automorphism and a
derivation. Via the Taylor series of the logarithm and of the exponential,

$\log(\sigma)=\sum_{n\geqslant
1}\frac{(-1)^{n+1}}{n}(\operatorname{Id}-\sigma)^{[n]}\text{ and
}\exp(\partial)=\sum_{n\geqslant 0}\frac{1}{n!}\partial^{[n]}$ (1)

(where $\varphi^{[n]}$ denotes the $n^{\text{th}}$ iterate of a map
$\varphi$), one obtains the fundamental relation between the group of
automorphisms and the Lie algebra of derivations of $k(\\!(t)\\!)$ (cf. [22,
Chapter 3]).

Fields of generalized power series $k(\\!(G)\\!)$ with coefficients in $k$ and
exponents in an arbitrary ordered abelian group $G$ are instrumental to the
valuation and model theory of fields (for example, when $G=\mathbb{Z}$, then
$k(\\!(G)\\!)$ is just the Laurent series field). In [19], the authors
described the group of valuation preserving automorphisms of $k(\\!(G)\\!)$,
as a semidirect product of four distinct factors.

Our initial motivation for the present work was to describe the Lie algebra of
derivations on $k(\\!(G)\\!)$, exploiting our knowledge of its group of
valuation preserving automorphisms. To this end, we cast the problem into the
following more general setting. Given a $k$-algebra $A$ (not necessarily
commutative), we study the interplay between the Lie algebra of its
derivations and the group of its automorphisms. Here, the essential issue is
to give appropriate conditions under which the infinite sums in (1) are indeed
well-defined. To deal with this issue, we explore a general notion of
summability in algebras, and of strongly linear maps (that is, linear maps
that commute with infinite sums). We show that there is a large class of
algebras in which the exponential and logarithm induce the desired
correspondence for strongly linear $\partial$ and $\sigma$.

In Section 1 we introduce the axiomatic notion of vector spaces and algebras
with a structure of formal summability, called summability spaces and
summability algebras (see Definitions 1.1 and 1.27). A related notion has been
presented independently in a categorical framework by Freni [7]. We show that
the algebra of strongly linear maps on a summability space inherits a natural
structure of summability algebra (Proposition 1.30). In Section 1.4, we first
recall the construction of the algebra $k\langle\\!\langle J\rangle\\!\rangle$
of formal series with coefficients in $k$ and non-commuting variables
$X_{j},j\in J$. We show in Proposition 1.34 that this algebra has a natural
structure of summability algebra.

In Section 2, we introduce the notion of summability algebras with evaluations
(see Definition 2.3). This allows any power series in $k\langle\\!\langle
J\rangle\\!\rangle$ to be evaluated in $A$ for each set $J$ (see 2.4). More
precisely, such an algebra $A$ is local (Proposition 2.8), and each summable
family $\mathbf{a}=(\mathbf{a}_{j})_{j\in J}$ ranging in its maximal ideal
$\mathfrak{m}$ defines a unique strongly linear algebra morphism
$\operatorname{ev}_{\mathbf{a}}\colon k\langle\\!\langle
J\rangle\\!\rangle\longrightarrow A$ that maps each variable $X_{j}$ to
$\mathbf{a}_{j}$.

In this paper, we will apply evaluation morphisms for the univariate and
bivariate cases only. In the univariate case (when $J$ is a singleton), we can
identify $k\langle\\!\langle J\rangle\\!\rangle$ with $k[\\![X]\\!]$ (the
algebra of formal power series in the variable $X$). In particular, the formal
power series

$\log(1+X):=\sum_{n\geqslant 1}\frac{(-1)^{n+1}}{n}X^{n}\text{ and
}\exp(X):=\sum_{n\geqslant 0}\frac{1}{n!}X^{n}$

can be evaluated at each $\varepsilon\in\mathfrak{m}$. Furthermore, the
relations

$\exp(\log(1+\varepsilon))=1+\varepsilon\ \text{ \ and }\
\log(\exp(\varepsilon))=\varepsilon$

follow (see Corollary 2.10) from evaluating the corresponding identities [25,
Section 1.7, Theorem 7.2] in $k[\\![X]\\!]$ at $\varepsilon$. In the bivariate
case, we likewise obtain (see Corollary 2.12) the Baker–Campbell–Hausdorff
formula (see [9, Section 1.3.2] or [25, Section 1.4.7])

$\log(\exp(\varepsilon)\cdot\exp(\delta))=\varepsilon+\delta+\frac{1}{2}[\varepsilon,\delta]+\frac{1}{12}([\varepsilon,[\varepsilon,\delta]]-[\delta,[\varepsilon,\delta]])+\cdots$

for $\varepsilon,\delta\in\mathfrak{m}$. Our first main result is a
correspondence between automorphisms and derivations (Theorem 2.14) for all
summability subalgebras (of strongly linear maps) with evaluations.

The purpose of Section 3 is to provide a large class of algebras to which
Theorem 2.14 applies. To this end, we introduce algebras of Noetherian series
in Section 3.3. Those include algebras of polynomials, algebras of formal
series in commuting or non-commuting variables, and fields of generalised
power series $k(\\!(G)\\!)$. Given an algebra $\mathbb{A}$ of Noetherian
series, we consider the algebra $k\operatorname{Id}_{\mathbb{A}}+\mathfrak{m}$
where $\mathfrak{m}$ is the closed ideal of contracting strongly linear maps
(see Definition 3.12). We show (Theorem 3.16) that this is an algebra with
evaluations. Our second main result Theorem 3.17 is that the exponential is a
bijection between the subset of $\mathfrak{m}$ consisting of derivations and
the subset of $\operatorname{Id}_{\mathbb{A}}+\mathfrak{m}$ consisting of
automorphisms of $\mathbb{A}$. In the case when $\mathbb{A}=k(\\!(G)\\!)$ is a
field of Hahn series, this group is one of the three factors in the
decomposition [19] of strongly linear valuation preserving automorphisms of
$k(\\!(G)\\!)$. It follows in particular (Corollary 3.20) that this group is
divisible and torsion-free. We derive from Theorem 3.17 our third and final
main result Theorem 3.19, which moreover takes into account the Lie structure
on the corresponding algebras. More precisely, we prove a formal analog of the
Lie homomorphism theorem (cf. [9, Theorem 3.7]) for contracting strongly
linear derivations and strongly linear automorphisms.

In Section 4, we focus on the case where $k$ is an ordered exponential field
[15], and we discuss the possibility of extending our correspondence to the
other two factors in the semidirect decomposition of the group of valuation
preserving automorphisms of $k(\\!(G)\\!)$.

### Conventions and notations

We denote by $\mathbb{N}$ the set of natural numbers _with_ $0$ and by
$\mathbb{N}^{\char 62\relax 0}$ _without_ $0$. The power set of a set $X$ is
denoted by $\mathcal{P}(X)$ and its subset consisting of all finite subsets of
$X$ is denoted by $\mathcal{P}_{\mathrm{fin}}(X)$.

Given sets $A,B$, a function $f\colon A\longrightarrow B$ is a subset of
$A\times B$ with the functional property. We often also identify the function
$f$ with the corresponding family $(f(a))_{a\in A}$. The set of functions from
$A$ to $B$ is denoted by $B^{A}$. If $B\subseteq C$, then $B^{A}\subseteq
C^{A}$. Note also that $A^{\emptyset}=\\{\emptyset\\}$.

Recall that an ordering on a set $\Omega$ is a binary relation $<$ on $\Omega$
such that for all $p,q,r\in\Omega$, we have

$p\nless p\text{ and }[(p<q\wedge q<r)\Longrightarrow p<r].$

We then say that $(\Omega,<)$ is an ordered set and for $p,q\in\Omega$, we
write $p\leqslant q$ if $p=q$ or $p<q$. If
$u\colon\mathbb{N}\longrightarrow\Omega$ is a sequence, then we say that $u$
is increasing (resp. strictly increasing) if for all $m,n\in\mathbb{N}$ with
$m<n$, we have $u(m)\leqslant u(n)$ (resp. $u(m)<u(n)$). A subsequence of $u$
is a sequence $v=u\circ\varphi\colon\mathbb{N}\longrightarrow\Omega$ where
$\varphi\colon\mathbb{N}\longrightarrow\mathbb{N}$ is strictly increasing.

If $I$ is a set and $(M,+,0)$ is a monoid (i.e. an associative, unital magma),
then the support of an $\mathbf{f}\in M^{I}$ is the subset

$\operatorname{supp}\mathbf{f}:=\\{i\in I:\mathbf{f}(i)\neq 0\\}$

of $I$. This object will be ubiquitous in the paper.

Throughout the paper, we fix a field $k$. All vector spaces and algebras
considered below are over $k$. All considered algebras are associative, but
not be necessarily unital, nor commutative. We will sometimes assume that $k$
has characteristic $0$. Given a vector space $V$ and a set $I$, the set
$V^{I}$ of maps $\mathbf{v}\colon I\longrightarrow V$ is equipped with its
natural vector space structure (pointwise operations). Note that the subset
$V^{(I)}$ of maps with finite support is a subspace of $V^{I}$. Given vector
spaces $V_{1},V_{2}$, we write $\operatorname{Lin}(V_{1},V_{2})$ for the
vector space of linear maps $V_{1}\longrightarrow V_{2}$, and we set
$\operatorname{Lin}(V_{1}):=\operatorname{Lin}(V_{1},V_{1})$. A _unital_
algebra $A$ is called local if one of the following equivalent assertions is
satisfied [20, Theorem 19.1]:

  * $\bullet$

$A$ has a unique maximal left ideal;

  * $\bullet$

$A$ has a unique maximal right ideal;

  * $\bullet$

the set $A\setminus U(A)$ of non-units in $A$ is an ideal of $A$.

Then the maximal left and right ideals are equal to $A\setminus U(A)$. Given
an algebra $(A,+,\cdot)$ and $a,b\in A$, we write

$[a,b]:=a\cdot b-b\cdot a\in A,$

and we recall that $(A,+,[\cdot,\cdot])$ is a Lie algebra.

##  1 Formal summability in algebras

###  1.1 Summability spaces

Freni [7] introduced a category $\Sigma\operatorname{Vect}$ whose objects are
vector spaces equipped with a notion of formal sums. We propose an axiomatic
description of such spaces, in the vein of [11, Section 6.2], that is more
tailored to our purposes. It was shown by Freni to be equivalent to his [7,
Proposition 3.27].

Let $V$ be a vector space. We generalise the notion of finite summation
operators

$\Sigma_{n}\colon V^{n}\longrightarrow V,\ (v_{0},\ldots,v_{n-1})\mapsto
v_{0}+\cdot\cdot\cdot+v_{n-1}$

to abstract sums $\Sigma_{I}\mathbf{v}\in V$ of families $\mathbf{v}\colon
I\longrightarrow V$ indexed by possibly infinite sets $I$. Not all families
$\mathbf{v}$ in the space $V^{I}$ can be summed in a consistent way: if $I$ is
infinite and there is a $v\in V\setminus\\{0\\}$, then the constant family
$(v)_{i\in I}$ cannot be summed. So we are to introduce axioms specifying the
intended properties of sets $\operatorname{dom}\Sigma_{I}\subseteq V^{I}$ of
summable families in conjunction with properties of operators $\Sigma_{I}$. It
is very convenient for application to be able to do this for arbitrary sets
$I$. This leads us to the following notion of summability spaces.

For a set $I$, let $\Sigma_{I}$ be a $k$-linear function

$\Sigma_{I}\colon\operatorname{dom}\Sigma_{I}\longrightarrow V$

whose domain $\operatorname{dom}\Sigma_{I}$ is a subspace of the vector space
$V^{I}$. Given sets $I,J$, consider the following axioms:

SS1

$V^{(I)}\subseteq\operatorname{dom}\Sigma_{I}$ and
$\Sigma_{I}\mathbf{v}=\sum_{i\in\operatorname{supp}\mathbf{v}}\mathbf{v}(i)$
for all $\mathbf{v}\in V^{(I)}$.

SS2

if $\varphi\colon I\longrightarrow J$ is bijective and
$\mathbf{v}\in\operatorname{dom}\Sigma_{J}$, then
$\mathbf{v}\circ\varphi\in\operatorname{dom}\Sigma_{I}$, and
$\Sigma_{I}(\mathbf{v}\circ\varphi)=\Sigma_{J}v$.

SS3

if $I=\bigsqcup_{j\in J}I_{j}$ and
$\mathbf{v}\in\operatorname{dom}\Sigma_{I}$, then writing
$\mathbf{v}_{j}:=\mathbf{v}\\!\\!\upharpoonright_{I_{j}}$ for each $j\in J$,
we have

SS3a

$\mathbf{v}_{j}\in\operatorname{dom}\Sigma_{I_{j}}$ for all $j\in J$,

SS3b

$(\Sigma_{I_{j}}\mathbf{v}_{j})_{j\in J}\in\operatorname{dom}\Sigma_{J}$, and

SS3c

$\Sigma_{I}\mathbf{v}=\Sigma_{J}\left((\Sigma_{I_{j}}\mathbf{v}_{j})_{j\in
J}\right)$.

SS4

if $I=I_{1}\sqcup I_{2}$ and
$(\mathbf{v},\mathbf{w})\in\operatorname{dom}\Sigma_{I_{1}}\times\operatorname{dom}\Sigma_{I_{2}}$,
then the function $(\mathbf{v}\sqcup\mathbf{w})\colon I\longrightarrow V$
given by $(\mathbf{v}\sqcup\mathbf{w})(i_{1}):=\mathbf{v}(i_{1})$ and
$(\mathbf{v}\sqcup\mathbf{w})(i_{2})=\mathbf{w}(i_{2})$ for all $i_{1}\in
I_{1}$ and $i_{2}\in I_{2}$ lies in $\operatorname{dom}\Sigma_{I}$.

UF

For all $\mathbf{v}\in\operatorname{dom}\Sigma_{I}$ and all families
$f=(f_{i})_{i\in I}$ of $k$-valued functions $f_{i}:X_{i}\longrightarrow k$
with _finite_ domains $X_{i}$, writing $If:=\\{(i,x):i\in I\wedge x\in
X_{i}\\}$, the family $f\mathbf{v}:=(f_{i}(x)\mathbf{v}(i))_{(i,x)\in If}$
lies in $\operatorname{dom}\Sigma_{If}$.

The reader can see that these axioms generalise various properties of finite
summation operators $\Sigma_{n}$ for $n\in\mathbb{N}$, including associativity
and commutativity (or invariance under reindexing) of the sum.

######  Definition 1.1.

We say that $\Sigma=(\Sigma_{I})_{I}$ (where $I$ ranges in the class of all
sets) is a summability structure on $V$, or that $(V,\Sigma)$ is a summability
space, if $(V,\Sigma)$ satisfies the axioms SS1 to SS4 above for all sets
$I,J$. If, moreover, axiom UF is satisfied for all $I$, then the summability
space $(V,\Sigma)$ is called ultrafinite.

######  Remark 1.2.

In the ultrafinite case, one can bound the cardinality of supports of summable
families by that of the underlying vector space. Thus we could consider
families $\Sigma$ indexed by $I$ in the powerset of $V$. Our choice of
indexing by all sets is for practical purposes. ∎

Throughout the rest of the section $(V,\Sigma)$ denotes a summability space.

######  Definition 1.3.

For any set $I$, any element of $\operatorname{dom}\Sigma_{I}$ is called a
summable family.

In the following, we establish some basic properties of summability spaces.

######  Notation 1.4.

We shall write $\sum_{i\in I}\mathbf{v}(i)$ for $\Sigma_{I}\mathbf{v}$.

######  Lemma 1.5.

Let $(V,\Sigma)$ be a summability space, let $I$, $I_{1}$, $I_{2}$ be sets and
let $f=(f_{i})_{i\in I}$ is a family of $k$-valued functions with finite
domains $X_{i}:=\operatorname{dom}f_{i}$. Then the following hold:

  1. (i)

If $I$ is finite and of the form $I=\\{i_{1},\ldots,i_{n}\\}$ for some
$n\in\mathbb{N}$, then $\operatorname{dom}\Sigma_{I}=V^{I}=V^{(I)}$ and for
any $\mathbf{v}\in V^{I}$ we have $\Sigma_{I}=\sum_{i\in
I}\mathbf{v}(i)=\sum_{j=1}^{n}\mathbf{v}(i_{j})$.

  2. (ii)

If $I=I_{1}\sqcup I_{2}$, then for any
$(\mathbf{v},\mathbf{w})\in\operatorname{dom}\Sigma_{I_{1}}\times\operatorname{dom}\Sigma_{I_{2}}$
we have
$\Sigma_{I}(\mathbf{v}\sqcup\mathbf{w})=\Sigma_{I_{1}}\mathbf{v}+\Sigma_{I_{2}}\mathbf{w}$.

  3. (iii)

If $(V,\Sigma)$ is ultrafinite, then for any $\mathbf{v}\in\Sigma_{I}$ also
$\big{(}(\sum_{x\in X_{i}}f_{i}(x))\mathbf{v}(i)\big{)}_{i\in I}\in\Sigma_{I}$
and

$\Sigma_{If}(f\mathbf{v})=\sum_{i\in I}\left(\sum_{x\in
X_{i}}f_{i}(x)\right)\mathbf{v}(i).$

Proof.

  1. (i)

This follows immediately from SS1.

  2. (ii)

Let $\mathbf{u}=(\mathbf{v}\sqcup\mathbf{w})$. Then
$\mathbf{u}_{1}=\mathbf{v}$ and $\mathbf{u}_{2}=\mathbf{w}$. By (i), SS3 and
SS4,

$\Sigma_{I}\mathbf{u}=\Sigma_{\\{1,2\\}}(\Sigma_{I_{j}}\mathbf{u}_{j})_{j\in\\{1,2\\}}=\Sigma_{I_{1}}\mathbf{u}_{1}+\Sigma_{I_{2}}\mathbf{u}_{2}=\Sigma_{I_{1}}\mathbf{v}+\Sigma_{I_{2}}\mathbf{w}.$

  3. (iii)

For any $i\in I$, let $I_{j}=\\{i\\}\times X_{i}$. Then $If=\bigsqcup_{i\in
I}I_{i}$. Applying SS3, UF and (i), we obtain

$\displaystyle\Sigma_{If}(f\mathbf{v})$
$\displaystyle=\Sigma_{I}\left((\Sigma_{I_{i}}(f_{i}(x)\mathbf{v}(i))_{(i,x)\in\\{i\\}\times
X_{i}})_{i\in I}\right)=\sum_{i\in I}\left(\sum_{x\in
X_{i}}f_{i}(x)\mathbf{v}(i)\right)$ $\displaystyle=\sum_{i\in
I}\left(\sum_{x\in X_{i}}f_{i}(x)\right)\mathbf{v}(i).$

$\Box$

We now show that any summability space satisfies a version of Dirichlet’s
rearrangement theorem.

######  Proposition 1.6.

Let $(V,\Sigma)$ be a summability space. Let $I,J$ be sets and let
$\mathbf{v}\in\operatorname{dom}\Sigma_{I\times J}$. Then for each $i_{0}\in
I$ and for each $j_{0}\in J$, we have
$\mathbf{v}(i_{0},\cdot)\in\operatorname{dom}\Sigma_{J}$ and
$\mathbf{v}(\cdot,j_{0})\in\operatorname{dom}\Sigma_{I}$. Moreover,
$\left(\sum_{j\in J}\mathbf{v}(i,j)\right)_{i\in
I}\in\operatorname{dom}\Sigma_{I}$ and $\left(\sum_{i\in
I}\mathbf{v}(i,j))\right._{j\in J}\in\operatorname{dom}\Sigma_{J}$, with

$\sum_{i\in I}\left(\sum_{j\in J}\mathbf{v}(i,j)\right)_{j\in
J}=\sum_{(i,j)\in I\times J}\mathbf{v}(i,j)=\sum_{j\in J}\left(\sum_{i\in
I}\mathbf{v}(i,j)\right)_{i\in I}.$

Proof.  Apply SS3 for $I\times J$ both with $(I\times J)_{j}:=I\times\\{j\\}$
for each $j\in J$ and with $(I\times J)_{i}:=\\{i\\}\times J$ for each $i\in
I$. $\Box$

Let $W\subseteq V$ be a subspace. Let
$\Sigma^{\prime}=(\Sigma^{\prime}_{I})_{I\text{ is a set}}$ be a family of
partial functions $\Sigma^{\prime}_{I}\\!\colon W^{I}\longrightharpoonup W$.
We say that $\Sigma^{\prime}$ is a restriction of $\Sigma$ if for each $I$,
the partial function $\Sigma_{I}$ extends $\Sigma^{\prime}_{I}$ (that is
$\operatorname{dom}\Sigma^{\prime}_{I}\subseteq\operatorname{dom}\Sigma_{I}$
and $\Sigma_{I}$ restricts to $\Sigma^{\prime}_{I}$ on
$\operatorname{dom}\Sigma^{\prime}_{I}$).

######  Definition 1.7.

Let $W\subseteq V$ be a subspace.

  1. (i)

We say that $W$ is a closed subspace if for any set $I$ and any
$\mathbf{w}\in\operatorname{dom}\Sigma_{I}\cap W^{I}$, we have
$\Sigma_{I}\mathbf{w}\in W$.

  2. (ii)

If $W$ is a closed subspace of $V$, then $\Sigma$ naturally restricts to a
summability structure $\mathbf{\Sigma^{W}}$ on $W$ by setting
$\operatorname{dom}\Sigma_{I}^{W}:=\operatorname{dom}\Sigma_{I}\cap
W^{I}\text{ and }\Sigma_{I}^{W}\mathbf{w}:=\Sigma_{I}\mathbf{w}$ for any set
$I$ and $\mathbf{w}\in\operatorname{dom}\Sigma^{W}_{I}$.

We leave it to the reader to check that $(W,\Sigma^{W})$ in Definition 1.7 is
indeed a summability space, and that $(W,\Sigma^{W})$ is ultrafinite if
$(V,\Sigma)$ is ultrafinite. We next give some simple examples of summability
spaces.

######  Example 1.8.

Any vector space $V$ has a summability structure called the minimal
summability structure, where for each set $I$, we have

$\operatorname{dom}\Sigma_{I}=V^{(I)}\text{ and }\forall\mathbf{v}\in
V^{(I)},\Sigma_{I}\mathbf{v}=\sum_{i\in\operatorname{supp}\mathbf{v}}\mathbf{v}(i).$

This structure is ultrafinite. ∎

######  Example 1.9.

Let $(\Gamma,<)$ be a non-empty linearly ordered set. Let
$\mathbf{H}_{\gamma\in\Gamma}k$ denote the Hahn product of the constant family
$(k)_{\gamma\in\Gamma}$, i.e. the vector space of functions $\Gamma\rightarrow
k$ with well-ordered support, under pointwise sum and scalar product. A
natural summability structure $\Sigma$ on $V:=\mathbf{H}_{\gamma\in\Gamma}k$
is obtained as follows. Given a set $I$, a function $\mathbf{v}\colon
I\rightarrow V$ is formally summable if the subset
$S_{\mathbf{v}}:=\bigcup\limits_{i\in I}\operatorname{supp}\mathbf{v}(i)$ of
$\Gamma$ is well-ordered, and for each $\gamma\in\Gamma$, the set
$I_{\mathbf{v},\gamma}:=\\{i\in I\ :\
\gamma\in\operatorname{supp}\mathbf{v}(i)\\}$ is finite. Define
$\operatorname{dom}\Sigma_{I}:=\\{\mathbf{v}\colon I\longrightarrow V\ :\
\mathbf{v}\ \text{is formally summable}\ \\}$ as the set of formally summable
families $I\longrightarrow V$. For
$\mathbf{v}\in\operatorname{dom}\Sigma_{I}$, we define $\Sigma_{I}\mathbf{v}$
to be the function

$\Gamma\longrightarrow k,\ \ \gamma\mapsto\sum\limits_{i\in
I_{\mathbf{v},\gamma}}\mathbf{v}(i),$

whose support is indeed well-ordered, as it is contained in $S_{\mathbf{v}}$.
This summability space is ultrafinite. ∎

######  Example 1.10.

Assume that $k=\mathbb{R}$ or $k=\mathbb{C}$. Let $\left(V,\left|\hskip
8.00003pt\right|\right)$ be a Banach space with absolute value $\left|\hskip
8.00003pt\right|\colon V\longrightarrow\mathbb{R}^{\geqslant 0}$. For any set
$I$, define $\operatorname{dom}\Sigma_{I}$ to be the set of families
$\mathbf{f}\colon I\longrightarrow V$ with countable or finite support, such
that given a bijection
$i\colon\lambda\longrightarrow\operatorname{supp}\mathbf{f}$ where
$\lambda\leqslant\omega$, the real-valued sequence
$\left(\sum_{p=0}^{n}|\mathbf{f}(i(p))|\right)_{n\in\lambda}$ converges. We
then define $\sum_{I}\mathbf{f}=\sum_{n\in\lambda}\mathbf{f}(i(n))$. Note that
this does not depend on the choice of bijection.

This summability structure is not ultrafinite in general. Indeed if $V$ is
non-trivial, then for any non-zero $v\in V$, the family
$\left(\frac{v}{n!}\right)_{n\in\mathbb{N}}$ is summable, whereas
$(vn!)_{n\in\mathbb{N}}$ is not. ∎

#### Bornological spaces

We now consider an important class of summability spaces called bornological
spaces (see also [7, Section 2.1]). We will appeal to this in Section 2 and
Section 3.

######  Definition 1.11.

Let $\Omega$ be a set. A bornology on $\Omega$ is a set $\mathcal{B}$ of
subsets of $\Omega$ which contains all finite subsets of $\Omega$ and which is
closed under finite unions and subsets, i.e. such that for all
$A,B\subseteq\Omega$, we have

$(A\in\mathcal{B}\wedge B\in\mathcal{B})\Longrightarrow A\cup
B\in\mathcal{B}\text{{\hskip 30.00005pt}and{\hskip 30.00005pt}$(A\subseteq
B\wedge B\in\mathcal{B})\Longrightarrow A\in\mathcal{B}$.}$

Given a bornology $\mathcal{B}$ on $\Omega$ and a vector space $V$, we write
$V^{[\mathcal{B}]}$ for the set of functions $\Omega\longrightarrow V$ whose
support lies in $\mathcal{B}$. Assume that $(V,\Sigma)$ is a summability space
and let $\Omega$ be a non-empty set. We define a structure of summability
space on $V^{[\mathcal{B}]}$ as follows. If $I$ is a set and $\mathbf{f}\colon
I\longrightarrow V^{[\mathcal{B}]}$ is a function, then we say that
$\mathbf{f}$ is $\mathcal{B}$-summable if

  * $\bullet$

the set $\bigcup_{i\in I}\operatorname{supp}\mathbf{f}(i)$ lies in
$\mathcal{B}$, and

  * $\bullet$

for all $p\in\Omega$, the family $(\mathbf{f}(i)(p))_{i\in I}$ is summable in
$(V,\Sigma)$.

If $\mathbf{f}$ is $\mathcal{B}$-summable, then we define
$\sum^{\mathcal{B}}_{I}\mathbf{f}\in V^{[\mathcal{B}]}$ to be the function
$p\mapsto\sum_{i\in I}\mathbf{f}(i)(p)$.

In the particular case when $\mathcal{B}=\mathcal{P}(\Omega)$, we have
$V^{[\mathcal{B}]}=V^{\Omega}$. We say that a family $\mathbf{f}\colon
I\longrightarrow V^{\Omega}$ is pointwise summable with respect to $\Sigma$ if
it is $\mathcal{P}(\Omega)$-summable, i.e. if each family
$(\mathbf{f}(i)(p))_{i\in I}$ for $p\in\Omega$ is summable in $(V,\Sigma)$.

######  Proposition 1.12.

Let $(V,\Sigma)$ be a summability space. Then
$(V^{[\mathcal{B}]},\Sigma^{\mathcal{B}})$ is a summability space. Moreover,
it is ultrafinite if $(V,\Sigma)$ is ultrafinite.

Proof.  Throughout the proof, we fix sets $I,J$ and a generic element
$p\in\Omega$.

Let $\mathbf{f},\mathbf{g}:I\longrightarrow V^{[\mathcal{B}]}$ be
$\mathcal{B}$-summable families and let $c\in k$. For all $i\in I$, we have

$\operatorname{supp}(\mathbf{f}(i)+c\mathbf{g}(i))\subseteq(\operatorname{supp}\mathbf{f}(i))\cup(\operatorname{supp}\mathbf{g}(i))$

so $\bigcup_{i\in
I}\operatorname{supp}((\mathbf{f}+c\mathbf{g})(i))\subseteq\left(\bigcup_{i\in
I}\operatorname{supp}\mathbf{f}(i)\right)\cup\left(\bigcup_{i\in
I}\operatorname{supp}\mathbf{g}(i)\right)$.

Since $\bigcup_{i\in I}\operatorname{supp}\mathbf{f}(i),\bigcup_{i\in
I}\operatorname{supp}\mathbf{g}(i)\in\mathcal{B}$ and $\mathcal{B}$ is closed
under finite unions we deduce that $\bigcup_{i\in
I}\operatorname{supp}((\mathbf{f}+c\mathbf{g})(i))$. Since
$\operatorname{dom}\Sigma_{I}$ is a vector subspace of $V^{I}$, the family
$(\mathbf{f}(i)(p)+c\mathbf{g}(i)(p))_{i\in I}$ is summable in $(V,\Sigma)$.
This shows that $\mathbf{f}+c\mathbf{g}$ is $\mathcal{B}$-summable. Moreover

$\sum_{I}(\mathbf{f}(i)(p)+c\mathbf{g}(i)(p))_{i\in
I}=\sum_{I}(\mathbf{f}(i)(p))_{i\in I}+c\sum_{I}(\mathbf{g}(i)(p))_{i\in I},$

so $\sum_{i\in I}(\mathbf{f}+c\mathbf{g})=\sum_{i\in I}\mathbf{f}+c\sum_{i\in
I}\mathbf{g}$.

If $\mathbf{f}$ has finite support, then $\bigcup_{i\in
I}\operatorname{supp}\mathbf{f}(i)$ is a finite union of sets in
$\mathcal{B}$, whence an element of $\mathcal{B}$. The family
$F=(\mathbf{f}(i)(p))_{i\in I}$ has finite support, so it is summable with sum
$\sum_{i\in\operatorname{supp}F}\mathbf{f}(i)(p)$. We deduce that $\mathbf{f}$
is $\mathcal{B}$-summable, with $\sum_{i\in
I}\mathbf{f}=\sum_{i\in\operatorname{supp}F}\mathbf{f}(i)\in\mathcal{B}$. So
SS1 holds for $\mathcal{B}$-summability.

Let $\varphi:J\longrightarrow I$ be a bijection. Then we have

$\bigcup_{j\in J}\operatorname{supp}\mathbf{f}(\varphi(j))=\bigcup_{i\in
I}\operatorname{supp}\mathbf{f}(i)$

which lies in $\mathcal{B}$. The family $(\mathbf{f}(\varphi(j))(p))_{j\in J}$
lies in $\operatorname{dom}\Sigma_{L}$ by SS2 in $(V,\Sigma)$, and we have

$\left(\sum_{j\in J}\mathbf{f}(\varphi(j))\right)(p)=\sum_{j\in
J}\mathbf{f}(\varphi(j))(p)=\sum_{i\in I}\mathbf{f}(i)(p)=\left(\sum_{i\in
I}\mathbf{f}(i)\right)(p).$

So $\sum_{J}\mathbf{f}\circ\varphi=\sum_{I}\mathbf{f}$, whence SS2 holds for
$\mathcal{B}$-summability.

Now assume that $I=\bigsqcup_{j\in J}I_{j}$. For $j\in J$, we have
$\bigcup_{i\in I_{j}}\operatorname{supp}\mathbf{f}(i)\subseteq\bigcup_{i\in
I}\operatorname{supp}\mathbf{f}(i)$ so $\bigcup_{i\in
I_{j}}\operatorname{supp}\mathbf{f}(i)\in\mathcal{B}$ by closedness of
$\mathcal{B}$ under subsets. The family $(\mathbf{f}(i)(p))_{i\in I_{j}}$ is
also summable by SS3a in $(\Sigma,V)$. So $(\mathbf{f}(i))_{i\in I_{j}}$ is
$\mathcal{B}$-summable. We set $\mathbf{f}_{j}:=\sum_{i\in
I_{j}}\mathbf{f}(i)$.

We have

$\bigcup_{j\in J}\operatorname{supp}\mathbf{f}_{j}\subseteq\bigcup_{j\in
J}\left(\bigcup_{i\in
I_{j}}\operatorname{supp}\mathbf{f}(i)\right)\subseteq\bigcup_{i\in
I}\operatorname{supp}\mathbf{f}(i),$

so $\bigcup_{j\in J}\operatorname{supp}\mathbf{f}_{l}\in\mathcal{B}$. By SS3b
and SS3c in $(V,\Sigma)$, the family $(\mathbf{f}_{j}(p))_{j\in J}$ is
summable with sum $\sum_{i\in I}\mathbf{f}(i)(p)$. It follows that
$(\mathbf{f}_{j})_{j\in J}$ is $\mathcal{B}$-summable with sum $\sum_{i\in
I}\mathbf{f}(i)$. So SS3 holds.

Let $I_{1},I_{2}$ be sets with $I=I_{1}\sqcup I_{2}$, and let
$\mathbf{f}_{1}:I_{1}\longrightarrow V^{[\mathcal{B}]}$ and
$\mathbf{f}_{2}:I_{2}\longrightarrow V^{[\mathcal{B}]}$ be
$\mathcal{B}$-summable. We have

$\bigcup_{i\in
I}\operatorname{supp}(\mathbf{f}_{1}\sqcup\mathbf{f}_{2})(i)=(\bigcup_{i\in
I_{1}}\operatorname{supp}\mathbf{f}_{1}(i))\cup(\bigcup_{i\in
I_{2}}\operatorname{supp}\mathbf{f}_{2}(i)),$

which lies in $\mathcal{B}$ since it is closed under unions. The family
$((\mathbf{f}_{1}\sqcup\mathbf{f}_{2})(i)(p))_{i\in
I}=((\mathbf{f}_{1}(i)(p))_{i\in I_{1}}\sqcup(\mathbf{f}_{2}(i)(p)))_{i\in
I_{2}}$ is summable in $(V,\Sigma)$ by SS4 in $(V,\Sigma)$, so
$\mathbf{f}_{1}\sqcup\mathbf{f}_{2}$ is $\mathcal{B}$-summable.

Assume finally that $(V,\Sigma)$ is ultrafinite. Let $(h_{i})_{i\in I}$ be a
family indexed by $I$ of $k$-valued functions $h_{i}$ with finite domains
$X_{i}$. We have

$\bigcup_{i\in I}\bigcup_{x\in
X_{i}}\operatorname{supp}h_{i}(x)\mathbf{f}(i)\subseteq\bigcup_{i\in
I}\operatorname{supp}\mathbf{f}(i),$

which lies in $\mathcal{B}$. By ultrafiniteness, the family
$(h_{i}(x)\mathbf{f}(i)(p))_{i\in I\wedge x\in X_{i}}$ is summable, so
$(h_{i}(x)\mathbf{f}(i))_{i\in I\wedge x\in X_{i}}$ is $\mathcal{B}$-summable.
So UF holds. $\Box$

We say that $(V^{[\mathcal{B}]},\Sigma^{\mathcal{B}})$ is a bornological
space. For $v\in V$ and $p\in\Omega$, write $\mathbbm{1}_{\\{p\\}}v$ for the
function $\Omega\longrightarrow V$ with $(\mathbbm{1}_{\\{p\\}}v)(q)=0$ if
$q\neq p$ and $(\mathbbm{1}_{\\{p\\}}v)(p)=v$. For $f\in V^{[\mathcal{B}]}$,
the family $(\mathbbm{1}_{\\{p\\}}f(p))_{p\in\Omega}$ is
$\mathcal{B}$-summable with sum $f$. So any element $f$ of $V^{[\mathcal{B}]}$
is a sum

$f=\sum_{p\in\Omega}\mathbbm{1}_{\\{p\\}}f(p),$ (2)

which can be considered as a formal series with coefficients in $V$ and with
support in $\mathcal{B}$.

######  Example 1.13.

On the Hahn space $\mathbf{H}_{n\in\mathbb{N}}k=k^{\mathbb{N}}$ of formal
power series, pointwise summability with respect to the minimal summability
structure on $k$ coincides withformal summability as described in 1.9.

∎

######  Example 1.14.

If $(G,+,<)$ is a totally ordered Abelian group, then the notion of Rayner
field family gives a bornology on $G$ contained in that of well-ordered
subsets of $G$. The resulting Rayner field [13] is a bornological space. These
fields were among those for which we sought to understand the derivation-
automorphism correspondence. Our results apply in particular to them. ∎

###  1.2 Strongly linear functions

Let $(V,\Sigma)$, $(V_{1},\Sigma_{1})$ and $(V_{2},\Sigma_{2})$ be summability
spaces.

######  Definition 1.15.

A function $\phi:V_{1}\longrightarrow V_{2}$ is said strongly linear if it is
linear, and if, for all sets $I$ and families
$\mathbf{v}\in\operatorname{dom}\Sigma_{1,I}$, the family
$\phi\circ\mathbf{v}$ lies in $\operatorname{dom}\Sigma_{2,I}$ and

$\Sigma_{2,I}(\phi\circ\mathbf{v})=\phi(\Sigma_{1,I}\mathbf{v}).$

We write $\operatorname{Lin}^{+}(V_{1},V_{2})$ for the set of strongly linear
functions $V_{1}\longrightarrow V_{2}$, and
$\operatorname{Lin}^{+}(V):=\operatorname{Lin}^{+}(V,V)$.

######  Example 1.16.

If $\Sigma_{1}$ is the minimal summability structure on $V_{1}$, then we have
$\operatorname{Lin}^{+}(V_{1},V_{2})=\operatorname{Lin}(V_{1},V_{2})$. ∎

######  Example 1.17.

Consider the Hahn space $V=\mathbf{H}_{\gamma\in\Gamma}k$ of 1.9. For all
order-preserving maps $u\colon\Gamma\longrightarrow\Gamma$, the function
$V\longrightarrow V\ \colon\ f\mapsto f\circ u$ is strongly linear. ∎

In general, linear maps between summability spaces that are not defined using
non-constructive methods often turn out to be strongly linear. As the next
example shows, using the axiom of choice allows one to define non-strongly
linear but linear functions.

######  Example 1.18.

Let $k^{\mathbb{N}}=k[\\![X]\\!]$ be the space of formal series with
coefficients in $k$, together with its summability structure of 1.9. Let $W$
be a complement to the subspace $k[X]$ of $k[\\![X]\\!]$ of functions
$\mathbb{N}\rightarrow k$ with finite support. Let $\mu$ be the unique linear
map $k[\\![X]\\!]\rightarrow k[\\![X]\\!]$ with $\mu(W)=\\{0\\}$ and
$\mu(X^{n})=(1+X)^{n}$ for each $n\in\mathbb{N}$. Then $\mu$ is not strongly
linear. Indeed, the family $(X^{n})_{n\in\mathbb{N}}$ is summable in
$k[\\![X]\\!]$ but $((1+X)^{n})_{n\in\mathbb{N}}$ is not summable, since $0$
lies in the support of each of its elements. ∎

######  Example 1.19.

If $V=\mathbf{H}_{\gamma\in\Gamma}k$ is the Hahn space of 1.9, then given a
strongly linear map $\mu\colon V\longrightarrow V$ and $a\in V$, we have

$\mu(a)=\mu(\sum\limits_{\gamma\in\Gamma}a(\gamma)\mathbbm{1}_{\gamma})=\sum\limits_{\gamma\in\Gamma}a(\gamma)\mu(\mathbbm{1}_{\gamma}).$

In particular, the function $\mu$ is determined by the family
$(\mu(\mathbbm{1}_{\gamma}))_{\gamma\in\Gamma}.$ ∎

######  Proposition 1.20.

$\operatorname{Lin}^{+}(V_{1},V_{2})$ is a subspace of
$\operatorname{Lin}(V_{1},V_{2})$,

Proof.  Let $\phi,\psi\in\operatorname{Lin}^{+}(V_{1},V_{2})$, $c\in k$, let
$I$ be a set and let $\mathbf{v}\in\operatorname{dom}\Sigma_{1,I}$. Since
$\operatorname{dom}\Sigma_{2,I}$ is a vector subspace of $V_{2}^{I}$ and
$\Sigma_{2,I}$ is a linear map, the family
$\phi\circ\mathbf{v}+c\psi\circ\mathbf{v}$ is summable with sum

$\Sigma_{2,I}(\phi\circ\mathbf{v})+c\Sigma_{2,I}(\psi\circ\mathbf{v})=\phi(\Sigma_{1,I}\mathbf{v})+c\psi(\Sigma_{2,I}\mathbf{v})=(\phi+c\psi)(\Sigma_{1,I}\mathbf{v}).$

This shows that $\phi+c\psi\in\operatorname{Lin}^{+}(V_{1},V_{2})$. $\Box$

######  Proposition 1.21.

$\operatorname{Lin}^{+}(V_{1})$ is a subalgebra of
$(\operatorname{Lin}(V_{1}),+,\circ)$.

Proof.  Let $\phi,\psi\in\operatorname{Lin}^{+}(V_{1})$, let $I$ be a set and
let $\mathbf{v}\in\operatorname{dom}\Sigma_{1,I}$. Then $\psi\circ\mathbf{v}$
is summable with sum
$\sum_{2,I}\psi\circ\mathbf{v}=\psi\left(\sum_{1,I}\mathbf{v}\right)$ by
strong linearity of $\psi$. So $(\phi\circ(\psi\circ\mathbf{v}))_{i\in I}$ is
summable by strong linearity of $\phi$, with sum
$\sum_{2,I}\phi\circ(\psi\circ\mathbf{v})=\phi\left(\psi\left(\sum_{1,I}\mathbf{v}\right)\right)$.
Since $\phi\circ\psi$ is linear, this shows that
$\phi\circ\psi\in\operatorname{Lin}^{+}(V_{1})$. $\Box$

######  Corollary 1.22.

$\operatorname{Lin}^{+}(V_{1})$ is a Lie subalgebra of
$\left(\operatorname{Lin}(V_{1}),+,\left[\hskip 10.00002pt\right],.\right)$.

We next equip $\operatorname{Lin}^{+}(V_{1},V_{2})$ with a structure of
summability space.

######  Definition 1.23.

Let $J$ be a set, and let
$\boldsymbol{\phi}:J\longrightarrow\operatorname{Lin}^{+}(V_{1},V_{2})$ be a
function.

  1. (i)

We say that $\boldsymbol{\phi}$ is $\operatorname{Lin}$-summable if for all
sets $I$ and all $\mathbf{v}\in\operatorname{dom}\Sigma_{1,I}$, the family
$\boldsymbol{\phi}(\mathbf{v}):=(\boldsymbol{\phi}(j)(\mathbf{v}(i)))_{(i,j)\in
I\times J}$ lies in $\operatorname{dom}\Sigma_{2,I\times J}$.

  2. (ii)

If $\boldsymbol{\phi}$ is $\operatorname{Lin}$-summable, then we define a
function $\sum_{j\in J}\boldsymbol{\phi}(j):V_{1}\longrightarrow V_{2}$ as
follows. For $v\in V_{1}$, define

$\left(\sum_{j\in
J}\boldsymbol{\phi}(j)\right)(v):=\Sigma_{2,J}(\boldsymbol{\phi}(j)(v))_{j\in
J}.$

######  Lemma 1.24.

If $J$ is a set and
$\boldsymbol{\phi}:J\longrightarrow\operatorname{Lin}^{+}(V_{1},V_{2})$ is
$\operatorname{Lin}$-summable, then its sum $\phi:=\sum_{j\in
J}\boldsymbol{\phi}(j)$ is strongly linear.

Proof.  Let $u_{0},v_{0}\in V$ and $c\in k$. The linearity of each
$\boldsymbol{\phi}(j)$ for $j\in J$ gives that
$\boldsymbol{\phi}(u_{0}+cv_{0})=\boldsymbol{\phi}(u_{0})+c\boldsymbol{\phi}(v_{0})$.
Now $\Sigma_{2,J}$ is linear, so
$\phi(u_{0}+cv_{0})=\phi(u_{0})+c\phi(v_{0})$, i.e. $\phi$ is linear.

We next prove that $\phi$ is strongly linear. Let $I$ be a set, let
$\mathbf{v}\in\operatorname{dom}\Sigma_{1,I}$ and set

$v_{1}:=\Sigma_{1,I}\mathbf{v}\in V_{1}.$

The family $\boldsymbol{\phi}(\mathbf{v})$ is summable. By Proposition 1.6,
both $\left(\sum_{j\in J}\boldsymbol{\phi}(j)(\mathbf{v}(i))\right)_{i\in I}$
and $\left(\sum_{i\in I}\boldsymbol{\phi}(j)(\mathbf{v}(i))\right)_{j\in J}$
are summable with

$\Sigma_{2,J}(\Sigma_{2,I}(\boldsymbol{\phi}(j)\circ\mathbf{v}))_{j\in
J}=\Sigma_{2,I}(\Sigma_{2,J}\boldsymbol{\phi}(\mathbf{v}(i)))_{i\in
I}=\Sigma_{2,I}(\phi\circ\mathbf{v}).$

By strong linearity of each $\boldsymbol{\phi}(j)$, we have
$\Sigma_{2,I}(\boldsymbol{\phi}(j)\circ\mathbf{v})=\boldsymbol{\phi}(j)(v_{0})$
for all $j\in J$, whence $\phi(v_{0})=\Sigma_{2,I}(\phi\circ\mathbf{v})$. This
shows that $\phi$ is strongly linear. $\Box$

For each set $I$, we define a function $\Sigma^{\operatorname{Lin}}_{I}$ as
follows. The domain of $\Sigma^{\operatorname{Lin}}_{I}$ is the set of
$\operatorname{Lin}$-summable families $V_{1}\longrightarrow V_{2}$ indexed by
$I$, and for such a family $\mathbf{\phi}$, we define
$\Sigma^{\operatorname{Lin}}_{I}=\Sigma_{i\in
I}\mathbf{\phi}(i)\in\operatorname{Lin}^{+}(V_{1},V_{2})$.

######  Proposition 1.25.

The structure
$(\operatorname{Lin}^{+}(V_{1},V_{2}),\Sigma^{\operatorname{Lin}})$ is a
summability space. Moreover, it is ultrafinite if $(V_{2},\Sigma_{2})$ is
ultrafinite.

Proof.  Let $J$ be a set, let $(\boldsymbol{\phi}(j))_{j\in J}$ and
$(\boldsymbol{\psi}(j))_{j\in J}$ be $\operatorname{Lin}$-summable, and let
$c\in k$. We also fix once and for all a set $I$ and a summable family
$\mathbf{v}\in\operatorname{dom}\Sigma_{1,I}$.

Note that for each set $L$ and $\operatorname{Lin}$-summable family
$\boldsymbol{\varphi}$ indexed by $L$, the function $\Sigma_{l\in
L}\boldsymbol{\varphi}(l)$ the pointwise sum
$\Sigma_{L}^{\mathcal{P}(V_{1})}\boldsymbol{\varphi}$ in the bornological
space $(V_{2})^{(\mathcal{P}(V_{1}))}$. So the equalities in SS1, SS2 and SS3c
are automatically satisfied, and we only need to show the summability of
families involved in the axioms SS1–SS4 and UF.

The family

$(\boldsymbol{\phi}(j)(\mathbf{v}(i))+c\boldsymbol{\psi}(j)(\mathbf{v}(i)))_{(i,j)\in
I\times J}$

is summable by SS2 in $(V_{2},\Sigma_{2})$ and definition of
$\operatorname{Lin}$-summability. So $\boldsymbol{\phi}+c\boldsymbol{\psi}$ is
$\operatorname{Lin}$-summable. The axiom SS1 follows trivially from the
validity of SS1 in $(V_{2},\Sigma_{2})$, and likewise SS4 follows from the
validity of SS4 in $(V_{2},\Sigma_{2})$.

If $J_{1},J_{2}$ are sets and $\varphi:J_{1}\longrightarrow J_{2}$ is a
bijection, then for all $\operatorname{Lin}$-summable
$\boldsymbol{\phi}:J_{2}\longrightarrow\operatorname{Lin}^{+}(V_{1},V_{2})$,
the function

$\displaystyle(\varphi,\operatorname{Id}):J_{1}\times I$
$\displaystyle\longrightarrow$ $\displaystyle J_{2}\times I$
$\displaystyle(j,i)$ $\displaystyle\longmapsto$ $\displaystyle(\varphi(j),i)$

is bijective, so by SS2 in $(V_{2},\Sigma)$, the family
$(\boldsymbol{\phi}(\varphi(j))(\mathbf{v}(i)))_{(j,i)\in J_{1}\times I}$ is
summable. Therefore $\boldsymbol{\phi}\circ\varphi$ is
$\operatorname{Lin}$-summable, i.e. SS2 holds.

Let us now show that SS3 holds. Suppose that $J=\bigsqcup_{l\in L}J_{l}$ for a
set $L$. For $l\in L$, we write $\boldsymbol{\phi}_{l}=\phi\upharpoonleft
J_{l}$. Note that

$I\times J=\bigsqcup_{l\in L}I\times J_{l}.$

It follows that and from SS3 in $(V_{2},\Sigma_{2})$ that each family
$\boldsymbol{\phi}_{l}(\mathbf{v})$ for $l\in L$ is summable in
$(V_{2},\Sigma_{2})$. So each $\boldsymbol{\phi}_{l}$ is
$\operatorname{Lin}$-summable. Write $\boldsymbol{\sigma}(l)=\sum_{j\in
J_{l}}\phi_{j}$ for each $l\in L$. We claim that
$\boldsymbol{\sigma}:=(\boldsymbol{\sigma}(l))_{l\in L}$ is
$\operatorname{Lin}$-summable. Indeed, the family

$\boldsymbol{\sigma}(\mathbf{v})=\left(\sum_{j\in
J_{l}}\phi_{j}(\mathbf{v}(i))\right)_{(i,l)\in I\times L}$

is summable by SS3 in $(V_{2},\Sigma_{2})$. Therefore SS3 holds.

Let us next prove that SS4 holds. Suppose that $J=J_{1}\sqcup J_{2}$, that
$\boldsymbol{\phi}_{1}:J_{1}\longrightarrow\operatorname{Lin}^{+}(V_{1},V_{2})$
and
$\boldsymbol{\phi}_{2}:J_{2}\longrightarrow\operatorname{Lin}^{+}(V_{1},V_{2})$
are $\operatorname{Lin}$-summable and write
$\boldsymbol{\phi}_{1}\sqcup\boldsymbol{\phi}_{2}=\boldsymbol{\varphi}$. Then
we have
$\boldsymbol{\varphi}(\mathbf{v})=\boldsymbol{\phi}_{1}(\mathbf{v})\sqcup\boldsymbol{\phi}_{2}(\mathbf{v})$,
so $\boldsymbol{\varphi}(\mathbf{v})$ is summable by SS4 in
$(V_{2},\Sigma_{2})$.

Assume now that $(V_{2},\Sigma)$ is ultrafinite. Let $(f(j))_{j\in J}$ be a
family of $k$-valued functions with finite domains
$\operatorname{dom}f_{j}=X_{j}$ for each $j\in J$. The family
$(f_{j}(x)\boldsymbol{\phi}(j)(\mathbf{v}(i)))_{(i,j)\in I\times J\wedge x\in
X_{j}}$ is summable by ultrafiniteness of $(V_{2},\Sigma_{2})$, so
$f\boldsymbol{\phi}$ is $\operatorname{Lin}$-summable. Thus UF holds. $\Box$

We now give a criterion for the summability of families of strongly linear
maps on bornological spaces.

######  Proposition 1.26.

Assume that $(V,\Sigma)$ is ultrafinite. Suppose that $k$ is equipped with the
minimal summability structure. Let $\Omega$ be a set and let $\mathcal{B}$ be
a bornology on $\Omega$. Let $J$ be a set and let $\boldsymbol{\phi}\colon
J\longrightarrow\operatorname{Lin}^{+}(k^{[\mathcal{B}]},V)$ be a function.
Then $\boldsymbol{\phi}$ is $\operatorname{Lin}$-summable if and only if for
all $S\in\mathcal{B}$, the family
$(\boldsymbol{\phi}(j)(\mathbbm{1}_{\\{p\\}}))_{(j,p)\in J\times S}$ is
summable in $(V,\Sigma)$.

Proof.  We need only prove the “if” direction of the equivalence. So let $I$
be a set and let $\mathbf{f}:I\longrightarrow k^{[\mathcal{B}]}$ be
$\mathcal{B}$-summable. Write

$I_{p}:=\\{i\in I:p\in\operatorname{supp}\mathbf{f}(i)\\}$

for each $p\in\Omega$, and set

$S:=\bigcup_{i\in I}\operatorname{supp}\mathbf{f}(i)\in\mathcal{B}.$

Since $S\in\mathcal{B}$, the family
$(\boldsymbol{\phi}(j)(\mathbbm{1}_{\\{p\\}}))_{j\in J\wedge p\in S}$ is
summable in $(V,\Sigma)$. For $p\in\Omega$, the family
$(\mathbf{f}(i)(p))_{i\in I}$ is summable in $k$, so $I_{p}$ is finite. We
deduce by ultrafiniteness that the family
$(\boldsymbol{\phi}(j)(\mathbbm{1}_{\\{p\\}}\mathbf{f}(i)(p)))_{j\in J\wedge
p\in\Omega\wedge i\in I_{p}}$ is summable, whence
$(\boldsymbol{\phi}(j)(\mathbbm{1}_{\\{p\\}}\mathbf{f}(i)(p)))_{j\in J\wedge
p\in\Omega\wedge i\in I}$ is summable by SS4 in $(V_{1},\Sigma_{1})$. It
follows from Proposition 1.6 that the family $\left(\sum_{p\in
S}\boldsymbol{\phi}(j)(\mathbbm{1}_{\\{p\\}}\mathbf{f}(i)(p))\right)_{j\in
J\wedge i\in I}$ is summable. Since each $\boldsymbol{\phi}(j)$ is strongly
linear, we have $(\boldsymbol{\phi}(j)(\mathbf{f}(i)))_{j\in J\wedge i\in
I}=\left(\sum_{p\in
S}\boldsymbol{\phi}(j)(\mathbbm{1}_{\\{p\\}}\mathbf{f}(i)(p))\right)_{j\in
J\wedge i\in I}$. So $(\boldsymbol{\phi}(j)(\mathbf{f}(i)))_{j\in J\wedge i\in
I}$ is summable, i.e. $\boldsymbol{\phi}$ is $\operatorname{Lin}$-summable.
$\Box$

###  1.3 Summability algebras

Let $(A,+,\cdot)$ be a algebra, where $(A,+,\cdot)$ is a possibly non-
commutative, possibly non-unital ring. Let $\Sigma$ be a summability structure
on the underlying vector space of $A$.

######  Definition 1.27.

We say that $(A,\Sigma)$ is a summability algebra if the following axiom is
satisfied.

$\mathbf{SA}$

For all sets $I,J$ and all
$(\mathbf{f},\mathbf{g})\in\operatorname{dom}\Sigma_{I}\times\operatorname{dom}\Sigma_{J}$,
the family
$\mathbf{f}\cdot\mathbf{g}:=(\mathbf{f}(i)\cdot\mathbf{g}(j))_{(i,j)\in
I\times J}$ lies in $\operatorname{dom}\Sigma_{I\times J}$, and we have

$\sum_{I\times
J}(\mathbf{f}\cdot\mathbf{g})=\left(\sum_{I}\mathbf{f}\right)\cdot\left(\sum_{J}\mathbf{g}\right).$

######  Proposition 1.28.

Let $(A,\Sigma)$ be a summability algebra. Then for $a\in A$, the left and
right product functions $\mathord{a\ \cdot}\>:A\longrightarrow A;b\mapsto
a\cdot b$ and $\mathord{\>\cdot\ a}\>:A\longrightarrow A;b\mapsto b\cdot a$
are strongly linear.

Proof.  By SA, for all summable families $(b_{i})_{i\in I}$ in $A$, the
families $(a\cdot b_{i})_{i\in I}$ and $(b_{i}\cdot a)_{i\in I}$ are summable,
with $\sum_{i\in I}a\cdot b_{i}=a\cdot\left(\sum_{i\in I}b_{i}\right)$ and
$\sum_{i\in I}b_{i}\cdot a=\left(\sum_{i\in I}b_{i}\right)\cdot a$. $\Box$

######  Example 1.29.

Let $(\Gamma,+,0,<)$ be a linearly ordered Abelian group, and consider the
summability space $A:=\mathbf{H}_{\gamma\in\Gamma}k$ of 1.9. Using the group
structure on $\Gamma$, one can define an algebra operation

$\forall f,g\in A,\forall\gamma\in\Gamma,(f\cdot
g)(\gamma):=\sum\limits_{\alpha+\beta=\gamma}f(\alpha)g(\beta).$

The definition is due to Hahn [8]. This algebra is a summability algebra. See
section 3 for more details. ∎

######  Proposition 1.30.

Let $(V,\Sigma)$ be a summability space. Then
$(\operatorname{Lin}^{+}(V),\Sigma^{\operatorname{Lin}})$ is a summability
algebra.

Proof.  By Proposition 1.25, we need only prove that SA holds in
$\operatorname{Lin}^{+}(V)$. Let $I$ and $J$ be sets, let
$\boldsymbol{\phi}\colon I\longrightarrow\operatorname{Lin}^{+}(V)$ and
$\boldsymbol{\psi}\colon J\longrightarrow\operatorname{Lin}^{+}(V)$ be
$\operatorname{Lin}$-summable with respective sums $\phi$ and $\psi$. Let $L$
be a set and let $\mathbf{v}\colon L\longrightarrow V$ be summable in
$(V,\Sigma)$.

Since $\boldsymbol{\psi}$ is $\operatorname{Lin}$-summable, the family
$(\boldsymbol{\psi}(j)(\mathbf{v}(l)))_{(j,l)\in J\times L}$ is summable in
$(V,\Sigma)$. Since $\boldsymbol{\phi}$ is $\operatorname{Lin}$-summable, the
family
$((\boldsymbol{\phi}(i)\circ\boldsymbol{\psi}(j))(\mathbf{v}(l)))_{(i,j,l)\in
I\times J\times L}$ is summable in $(V,\Sigma)$. This shows that
$(\boldsymbol{\phi}(i)\circ\boldsymbol{\psi}(j))_{(i,j)\in I\times J}$ is
$\operatorname{Lin}$-summable. Let $v_{0}\in V$. As above, the family

$F:=((\boldsymbol{\phi}(i)\circ\boldsymbol{\psi}(j))(v_{0}))_{(i,j,l)\in
I\times J}$

is summable in $(V,\Sigma)$. We have

$\displaystyle(\phi\circ\psi)(v_{0})$ $\displaystyle=\left(\sum_{i\in
I}\boldsymbol{\phi}(i)\right)\left(\sum_{j\in
J}\boldsymbol{\psi}(j)(v_{0})\right)$ $\displaystyle=\sum_{i\in
I}\boldsymbol{\phi}(i)\left(\sum_{j\in J}\boldsymbol{\psi}(j)(v_{0})\right)$
$\displaystyle=\sum_{i\in I}\sum_{j\in
J}\boldsymbol{\phi}(i)(\boldsymbol{\psi}(j)(v_{0}))$ (each
$\boldsymbol{\phi}(i)$ is strongly linear) $\displaystyle=\sum_{(i,j)\in
I\times J}\boldsymbol{\phi}(i)(\boldsymbol{\psi}(j)(v_{0})).$ (Proposition
1.6)

So $\phi\circ\psi=\sum_{I\times
J}\boldsymbol{\phi}(i)\circ\boldsymbol{\psi}(j)$. Therefore SA holds in
$\operatorname{Lin}^{+}(V)$. $\Box$

A closed ideal of a summability algebra $(A,\Sigma)$ is a _two-sided_ ideal of
$A$ which is also a closed subspace of $(A,\Sigma)$.

###  1.4 Algebras of formal power series

Let $J$ be a set. Write

$J^{\star}:=\bigcup_{n\in\mathbb{N}}J^{n}$

where $J^{0}=\\{\emptyset\\}$. We see elements of $J^{\star}$ as finite words
with letters in $J$.

For $m,n\in\mathbb{N}$, if
$\beta=(\beta_{1},\ldots,\beta_{m}),\gamma=(\gamma_{1},\ldots,\gamma_{n})\in
J^{\star}$, we define

$\beta\gamma:=(\beta_{1},\ldots,\beta_{m},\gamma_{1},\ldots,\gamma_{n})\in
J^{m+n}\subseteq J^{\star},$

where it is implied that $\emptyset\theta=\theta\emptyset=\theta$ for all
$\theta\in J^{\star}$. This concatenation operation endows $J^{\star}$ with a
structure of cancellative monoid, with the important property that given
$\theta\in J^{\star}$, the set

$\\{(\beta,\gamma)\in J^{\star}\times J^{\star}:\theta=\beta\gamma\\}$ (3)

is finite (it has exactly $n+1$ elements when $\theta\in J^{n}$).

The following construction, also considered in [22, Chapter 0, p 17], is a
particular case of Bourbaki’s notion of total algebra [3, Chapter III, Section
2.10].

######  Proposition 1.31.

The vector space $k^{J^{\star}}$ is a unital algebra under the Cauchy product

$(P\cdot Q)(\theta):=\sum_{\theta=\beta\gamma}P(\beta)Q(\gamma),$ (4)

for any $P,Q\in k^{J^{\star}}$ and any $\theta\in J^{\star}$.

######  Notation 1.32.

We write $k\langle\\!\langle J\rangle\\!\rangle:=k^{J^{\star}}$.

We recall some properties of $k\langle\\!\langle J\rangle\\!\rangle$. For
$P\in k\langle\\!\langle J\rangle\\!\rangle$ and $n\in\mathbb{N}$, we write

$\operatorname{supp}_{n}P:=(\operatorname{supp}P)\cap J^{n}=\\{\theta\in
J^{n}:P(\theta)\neq\emptyset\\}.$

Note that we have

$\operatorname{supp}_{n}P\cdot
Q\subseteq\bigcup_{m+p=n}(\operatorname{supp}_{m}P)\
(\operatorname{supp}_{p}Q),$ (5)

where $AB=\\{ab:(a,b)\in A\times B\\}$ for all subsets $A,B\subseteq
J^{\star}$. Concatenation of subsets of $J^{\star}$ is associative. An easy
induction gives:

######  Lemma 1.33.

For $m,n\in\mathbb{N}$ and $P_{1},\ldots,P_{n}\in k\langle\\!\langle
J\rangle\\!\rangle$, we have

$\operatorname{supp}_{m}P_{1}\cdots
P_{n}\subseteq\bigcup_{m_{1}+\cdots+m_{n}=m}(\operatorname{supp}_{m_{1}}P_{1})\cdots(\operatorname{supp}_{m_{n}}P_{n}).$

The set

$k\langle\\!\langle J\rangle\\!\rangle_{0}:=\\{P\in k\langle\\!\langle
J\rangle\\!\rangle:\emptyset\not\in\operatorname{supp}P\\}=\\{P-P(\emptyset):P\in
k\langle\\!\langle J\rangle\\!\rangle\\}$

is a two-sided ideal in $k\langle\\!\langle J\rangle\\!\rangle$.

Given $\theta\in J^{\star}$, we write $X_{\theta}$ for the function
$\mathbbm{1}_{\\{\theta\\}}\colon J^{\star}\longrightarrow k$ with support
$\\{\theta\\}$ and $X_{\theta}(\theta)=1$. So $X_{\emptyset}=1$, and writing
$\theta=(\theta_{1},\ldots,\theta_{n})$, we have

$X_{\theta}=X_{\theta_{1}}\cdots X_{\theta_{n}}.$

We will write $k\langle\\!\langle m\rangle\\!\rangle:=k\langle\\!\langle
J\rangle\\!\rangle$ for $m\in\mathbb{N}$ and $J=\\{0,\ldots,m-1\\}$. Note that
$k\langle\\!\langle 0\rangle\\!\rangle=k$ and that $k\langle\\!\langle
1\rangle\\!\rangle$ is the commutative algebra of power series in one variable
$X_{0}$ and with coefficients in $k$.

By Proposition 1.12, pointwise summability with respect to the minimal
summability structure on $k$ gives an ultrafinite summability structure on
$k\langle\\!\langle J\rangle\\!\rangle=k^{J^{\star}}$, which we denote by
$\Sigma$. Recall that a family $\mathbf{P}\colon I\longrightarrow
k\langle\\!\langle J\rangle\\!\rangle$ is pointwise summable here if and only
if for all $\theta\in J^{\star}$, the family $(\mathbf{P}(i)(\theta))_{i\in
I}$ has finite support, i.e. if and only if the set

$I_{\theta}:=\\{i\in I:\theta\in\operatorname{supp}\mathbf{P}(i)\\}$

is finite. As in (2), the series representation of a $P\in k\langle\\!\langle
J\rangle\\!\rangle$ is

$P=\sum_{\theta\in J^{\star}}P(\theta)X_{\theta}.$

######  Proposition 1.34.

The structure $(k\langle\\!\langle J\rangle\\!\rangle,\Sigma)$ is a
summability algebra.

Proof.  We need only show that SA holds. Let $I$ and $L$ be sets, let
$\mathbf{P}\colon I\longrightarrow k\langle\\!\langle J\rangle\\!$ and
$\mathbf{Q}\colon L\longrightarrow k\langle\\!\langle J\rangle\\!\rangle$ be
pointwise summable. Let $n\in\mathbb{N}$ and $\theta\in J^{n}$. We have

$\displaystyle(I\times L)_{\theta}$ $\displaystyle=$ $\displaystyle\\{(i,l)\in
I\times L:\theta\in\operatorname{supp}\mathbf{P}(i)\cdot\mathbf{Q}(j)\\}$
$\displaystyle=$ $\displaystyle\\{(i,l)\in I\times L:\exists\gamma,\beta\in
J^{\star},i\in I_{\beta}\wedge l\in L_{\gamma}\\}$

where for $\beta,\gamma\in J^{\star}$ the sets $I_{\beta}:=\\{i\in
I:\beta\in\operatorname{supp}\mathbf{P}(i)\\}$ and $L_{\gamma}:=\\{l\in
L:\gamma\in\operatorname{supp}\mathbf{Q}(l)\\}$ are finite. Since
$\\{(\beta,\gamma)\in J^{\star}\times J^{\star}:\beta:\gamma=\theta\\}$ is
finite, we deduce that $(I\times L)_{\theta}$ is finite. So
$(\mathbf{P}(i)\mathbf{Q}(l))_{(i,l)\in I\times L}$ is pointwise summable.

For $n\in\mathbb{N}$ and $\theta\in J^{n}$, the previous arguments give

$\displaystyle\left(\sum_{(i,l)\in I\times
L}\mathbf{P}(i)\cdot\mathbf{Q}(l)\right)(\theta)$ $\displaystyle=$
$\displaystyle\sum_{\beta:\gamma=\theta}\sum_{(i,l)\in I_{\beta}\times
L_{\gamma}}\mathbf{P}(i)(\beta)\mathbf{Q}(l)(\gamma)$ $\displaystyle=$
$\displaystyle\sum_{\beta:\gamma=\theta}\left(\sum_{i\in
I_{\beta}}\mathbf{P}(i)(\beta)\right)\left(\sum_{l\in
L_{\gamma}}\mathbf{Q}(l)(\gamma)\right)$ $\displaystyle=$
$\displaystyle\sum_{\beta:\gamma=\theta}\left(\sum_{i\in
I}\mathbf{P}(i)\right)(\beta)\left(\sum_{l\in L}\mathbf{Q}(l)\right)(\gamma)$
$\displaystyle=$ $\displaystyle\left(\left(\sum_{i\in
I}\mathbf{P}(i)\right)\cdot\left(\sum_{l\in
L}\mathbf{Q}(l)\right)\right)(\theta).$

This shows that $\sum_{(i,l)\in I\times
L}\mathbf{P}(i)\cdot\mathbf{Q}(l)=\left(\sum_{i\in
I}\mathbf{P}(i)\right)\cdot\left(\sum_{l\in L}\mathbf{Q}(l)\right)$, hence
that SA holds. $\Box$

##  2 Derivations and endomorphisms

###  2.1 Strongly linear derivations

Let $(A,\Sigma)$ be a summability algebra. We write
$\operatorname{End}^{+}(A)$ for the set of strongly linear endomorphisms of
algebra of $A$. Note by Proposition 1.21 that $\operatorname{End}^{+}(A)$ is
closed under composition.

######  Definition 2.1.

A strongly linear derivation on $A$ is a strongly linear function
$\partial:A\longrightarrow A$ which satisfies the Leibniz product rule

$\forall a,b\in A,\partial(a\cdot b)=\partial(a)\cdot b+a\cdot\partial(b).$

We write $\operatorname{Der}^{+}(A)$ for the set of strong derivations on $A$.

It follows from Proposition 1.21 that
$\left(\operatorname{Der}^{+}(A),+,\left[\hskip 11.99998pt\right],.\right)$ is
a Lie algebra.

######  Proposition 2.2.

$\operatorname{Der}^{+}(A)$ is a closed subspace of
$\operatorname{Lin}^{+}(A)$.

Proof.  Let $I$ be a set, let $(\partial_{i})_{i\in I}$ be a family in
$\operatorname{Der}^{+}(A)$ which is $\operatorname{Lin}$-summable, and write
$\partial:=\sum_{i\in I}\partial_{i}$. Let $a,b\in A$. We have

$\displaystyle\partial(a\cdot b)$ $\displaystyle=\sum_{i\in
I}\partial_{i}(a\cdot b)$ $\displaystyle=\sum_{i\in I}(\partial_{i}(a)\cdot
b+a\cdot\partial_{i}(b))$ $\displaystyle=\sum_{i\in I}\partial_{i}(a)\cdot
b+\sum_{i\in I}a\cdot\partial_{i}(b)$ $\displaystyle=\left(\sum_{i\in
I}\partial_{i}(a)\right)\cdot b+a\cdot\left(\sum_{i\in
I}\partial_{i}(b)\right)$ (SA) $\displaystyle=\partial(a)\cdot
b+a\cdot\partial(b).$

So $\partial\in\operatorname{Der}^{+}(A)$. $\Box$

###  2.2 Evaluating formal power series

Given a set $J$, the summability algebra $k\langle\\!\langle
J\rangle\\!\rangle$ extends the completion $k\left\langle J\right\rangle$ of
the free associative algebra on $J$. It is well-known [22, Chapter 0, p 17–18]
that elements of $k\langle J\rangle$ can be evaluated at tuples of elements in
its maximal ideal. This allows for the development of a formal Lie
correspondence for formal power series, based on the evaluation of the Taylor
series of the exponential and logarithm.

It is very convenient to extend these results to more general summability
algebras. Then one can see $k\langle\\!\langle J\rangle\\!\rangle$ as a
universal and free summability algebra, acting by evaluation on summability
algebras, so that universal identities (such as
$\exp\circ\log=\operatorname{id}$) that can be stated in summability algebras
could be proved once in $k\langle\\!\langle J\rangle\\!\rangle$ and then
obtained in general by evaluating into summability algebras.

######  Definition 2.3.

Let $(A,\Sigma)$ be a unital ultrafinite summability algebra. Then we say that
$(A,\Sigma)$ has evaluations if $A$ is of the form $A=k+\mathfrak{m}$ where
$\mathfrak{m}$ is a closed ideal, and for all sets $J$, all
$\mathbf{f}\in\operatorname{dom}\Sigma_{J}^{\mathfrak{m}}$, the family
$(\mathbf{f}(\theta_{1})\cdots\mathbf{f}(\theta_{n}))_{\theta=(\theta_{1},\ldots,\theta_{n})\in
J^{\star}}$ is summable in $(A,\Sigma)$.

######  Remark 2.4.

Let us justify this definition.

Given a summability algebra $A$ and a family $\mathbf{f}\colon
J\longrightarrow A$, the evaluation $\operatorname{ev}_{\mathbf{f}}(P)$ of a
formal power series $P\in k\langle\\!\langle J\rangle\\!\rangle$ at
$\mathbf{f}$ ought to be the sum of the family
$(P(j_{1},...,j_{n})\mathbf{f}(j_{1})\cdots\mathbf{f}(j_{n}))_{(j_{1},...,j_{n})\in
J^{\star}}$. Indeed, if all such families are summable, then
$\operatorname{ev}_{\mathbf{f}}$ will be the only strongly linear morphism of
algebras between $k\langle\\!\langle J\rangle\\!\rangle$ and $A$ that sends
$X_{j}$ to $\mathbf{f}(j)$ for each $j\in J$.

For $P=\sum\limits_{j\in J}X_{j}$, this entails that $\mathbf{f}$ be summable.
Furthermore, if $\mathbf{f}(j)\in k^{\times}$ for a certain $j\in J$, then for
$P=\sum\limits_{k\in\mathbb{N}}(\mathbf{f}(j))^{-k}X^{k}_{j}$, we see that the
family
$(P(j_{1},...,j_{n})\mathbf{f}(j_{1})\cdots\mathbf{f}(j_{n}))_{(j_{1},...,j_{n})\in
J^{\star}}$ has $(1)_{\theta\in\\{j\\}^{\star}}$ as a subfamily, hence is not
summable. So if evaluations are to be defined in the case when
$A=k\langle\\!\langle J\rangle\\!\rangle$, then $\mathbf{f}$ should range in
the maximal ideal $k\langle\\!\langle J\rangle\\!\rangle_{0}$ of
$k\langle\\!\langle J\rangle\\!\rangle$.

In general, writing $\mathfrak{m}$ for the set of elements
$\operatorname{ev}_{\mathbf{a}}(P)\in A$ where $J$ ranges among all sets, $P$
ranges among all elements in $k\langle\\!\langle J\rangle\\!\rangle_{0}$ and
$\mathbf{f}$ ranges among summable families for which
$(P(j_{1},...,j_{n})\mathbf{f}(j_{1})\cdots\mathbf{f}(j_{n}))_{(j_{1},...,j_{n})\in
J^{\star}}$ is summable, then $\mathfrak{m}$ is a two-sided ideal of the
algebra $A^{\prime}:=k+\mathfrak{m}$.

Imposing furthermore that $\mathfrak{m}$ be a closed subspace of $A$, we might
as well take $A=A^{\prime}$ and thus work with summability algebras of the
form $A=k+\mathfrak{m}$ for a two-sided ideal $\mathfrak{m}$ of $A$. ∎

We will see (Proposition 2.8) that such a summaility algebra $k+\mathfrak{m}$
is local with maximal ideal $\mathfrak{m}$. Let $J$ be set. Let us check that
$k\langle\\!\langle J\rangle\\!\rangle=k+k\langle\\!\langle
J\rangle\\!\rangle_{0}$ itself has evaluations.

######  Proposition 2.5.

The set $k\langle\\!\langle J\rangle\\!\rangle_{0}$ is a closed ideal of
$k\langle\\!\langle J\rangle\\!\rangle$, and $(k\langle\\!\langle
J\rangle\\!\rangle,\Sigma)$ has evaluations.

Proof.  Let $\mathbf{Q}\colon I\longrightarrow k\langle\\!\langle
J\rangle\\!\rangle_{0}$ be pointwise summable. We have $\left(\sum_{i\in
I}\mathbf{Q}(i)\right)(\emptyset)=\sum_{i\in\emptyset}\mathbf{Q}(i)(\emptyset)=0$,
so $\sum_{i\in I}\mathbf{Q}(i)\in k\langle\\!\langle J\rangle\\!\rangle_{0}$,
which is thus a closed ideal.

We next want to show that the family
$(\mathbf{Q}(i_{1})\cdots\mathbf{Q}(i_{n}))_{(i_{1},\ldots,i_{n})\in
I^{\star}}$ is pointwise summable. Let $m\in\mathbb{N}$ and
$\theta=(\theta_{1},\ldots,\theta_{m})\in J^{m}$. Writing $I_{\beta}:=\\{i\in
I:\beta\in\operatorname{supp}\mathbf{Q}(i)\\}$ for all $\beta\in J^{\star}$,
we have

$\displaystyle I^{\star}_{\theta}$ $\displaystyle=$ $\displaystyle\\{i\in
I^{n}:n\in\mathbb{N}\wedge\theta\in\operatorname{supp}_{m}\mathbf{Q}(i_{1})\cdots\mathbf{Q}(i_{n})\\}$
$\displaystyle\subseteq$
$\displaystyle\bigcup_{n\in\mathbb{N}}\bigcup_{\beta_{1}\cdots\beta_{n}=\theta}I_{\beta_{1}}\times\cdots\times
I_{\beta_{n}},$

Since each $\mathbf{Q}(i)$ lies in $k\langle\\!\langle
J\rangle\\!\rangle_{0}$, we have
$\operatorname{supp}_{0}\mathbf{Q}(i)=\emptyset$ for all $i\in I$, so we have
in fact

$I^{\star}_{\theta}\subseteq\bigcup_{n\leqslant
m}\bigcup_{\beta_{1}\cdots\beta_{n}=\theta}I_{\beta_{1}}\times\cdots\times
I_{\beta_{n}}.$

Now since each set
$X_{n}:=\\{(\beta_{1},\ldots,\beta_{n})\in(J^{\star})^{n}:\beta_{1}\cdots\beta_{n}=\theta\\}$
is finite and each $I_{\beta},\beta\in J^{\star}$ is finite, we deduce that
$I^{\star}_{\theta}$ is finite. Therefore the family
$(\mathbf{Q}(i_{1})\cdots\mathbf{Q}(i_{n}))_{i=(i_{1},\ldots,i_{n})\in
I^{\star}}$ is pointwise summable. $\Box$

If $(A,\mathfrak{m},\Sigma)$ are as in Definition 2.3, then for all sets $J$,
all $\mathbf{f}\in\operatorname{dom}\Sigma_{J}^{\mathfrak{m}}$ and all $P\in
k\langle\\!\langle J\rangle\\!\rangle$, we define the evaluation of $P$ at
$\mathbf{f}$ as

$\operatorname{ev}_{\mathbf{f}}(P):=\sum_{\theta=(\theta_{1},\ldots,\theta_{n})\in
J^{\star}}P(\theta)\mathbf{f}(\theta_{1})\cdots\mathbf{f}(\theta_{n})\in A.$

If $J=\\{0,\ldots,m-1\\}$ for an $m\in\mathbb{N}$, then we simply write
$\operatorname{ev}_{\mathbf{f}(0),\ldots,\mathbf{f}(m-1)}(P):=\operatorname{ev}_{\mathbf{f}}(P)$.

######  Theorem 2.6.

Assume that $A=k+\mathfrak{m}$ has evaluations. For all sets $J$ and all
$\mathbf{f}\in\operatorname{dom}\Sigma_{J}^{\mathfrak{m}}$, the evaluation map
$\operatorname{ev}_{\mathbf{f}}:k\langle\\!\langle
J\rangle\\!\rangle\longrightarrow A$ is a strongly linear morphism of
algebras.

Proof.  That $\operatorname{ev}_{\mathbf{f}}$ is a linear map is a direct
consequence of the fact that
$\Sigma_{\\{(n,\theta):n\in\mathbb{N}\wedge\theta\in J^{n}\\}}^{\mathfrak{m}}$
is linear. For $P,Q\in k\langle\\!\langle J\rangle\\!\rangle$, SA gives

$\displaystyle\operatorname{ev}_{\mathbf{f}}(P)\cdot\operatorname{ev}_{\mathbf{f}}(Q)$
$\displaystyle=$ $\displaystyle\sum_{m,p\in\mathbb{N}\wedge\beta\in
J^{m}\wedge\gamma\in
J^{p}}P(\beta)Q(\gamma)\mathbf{f}(\beta_{1})\cdots\mathbf{f}(\beta_{m})\cdot\mathbf{f}(\gamma_{1})\cdots\mathbf{f}(\gamma_{p})$
$\displaystyle=$ $\displaystyle\sum_{n\in\mathbb{N}\wedge\theta\in
J^{n}}(P\cdot Q)(\theta)\mathbf{f}(\theta_{1})\cdots\mathbf{f}(\theta_{n})$
$\displaystyle=$ $\displaystyle\operatorname{ev}_{\mathbf{f}}(P\cdot Q).$

So $\operatorname{ev}_{\mathbf{f}}$ is a morphism of algebras.

Let $\mathbf{P}\colon I\longrightarrow k\langle\\!\langle J\rangle\\!\rangle$
be pointwise summable and set $P:=\sum_{i\in I}\mathbf{P}(i)$. Consider the
element

$P^{\prime}:=\sum_{\theta\in J^{\star}}X_{\theta}$

of $k\langle\\!\langle J\rangle\\!\rangle$. Since $(A,\Sigma)$ has
evaluations, we have a pointwise summable family
$(P^{\prime}(\theta)\mathbf{f}(\theta_{1})\cdots\mathbf{f}(\theta_{n}))_{\theta=(\theta_{1},\ldots,\theta_{n})\in
J^{\star}}$. Consider the family of finite subsets of $k$

$(C_{(n,\theta)})_{\theta=(\theta_{1},\ldots,\theta_{n})\in
J^{\star}}=(\\{\mathbf{P}(i)(\theta):\mathbf{P}(i)(\theta)\neq
0\\}\\})_{\theta=(\theta_{1},\ldots,\theta_{n})\in J^{\star}}.$

Recall that $(A,\Sigma)$ is ultrafinite, so
$(cP^{\prime}(\theta)\mathbf{f}(\theta_{1})\cdots\mathbf{f}(\theta_{n}))_{\theta=(\theta_{1},\ldots,\theta_{n})\in
J^{\star}\wedge c\in C_{(n,\theta)}}$ is summable in $(A,\Sigma)$. By SS2, so
is the family

$\mathbf{g}:=(\mathbf{P}(i)(\theta)\mathbf{f}(\theta_{1})\cdots\mathbf{f}(\theta_{n}))_{\theta=(\theta_{1},\ldots,\theta_{n})\in
J^{\star}\wedge i\in I_{\theta}}.$

By SS4, the family

$(\operatorname{ev}_{\mathbf{f}}(\mathbf{P}(i)))_{i\in
I}=(\mathbf{P}(i)(\theta)\mathbf{f}(\theta_{1})\cdots\mathbf{f}(\theta_{n}))_{\theta=(\theta_{1},\ldots,\theta_{n})\in
J^{\star}\wedge i\in I}$

is summable as the union of the families $\mathbf{g}$ and
$(0)_{\theta=(\theta_{1},\ldots,\theta_{n})\in J^{\star}\wedge i\in I\setminus
I_{\theta}}$. Again by ultrafiniteness of $(A,\Sigma)$, we have

$\sum_{i\in
I}\operatorname{ev}_{\mathbf{f}}(\mathbf{P}(i))=\sum_{n\in\mathbb{N}\wedge\theta=(\theta_{1},...,\theta_{n})}\left(\sum_{i\in
I_{\theta}}\mathbf{P}(i)(\theta)\right)\mathbf{f}(\theta_{1})\cdots\mathbf{f}(\theta_{n})=\operatorname{ev}_{\mathbf{f}}(P).$

This shows that $\operatorname{ev}_{\mathbf{f}}$ is strongly linear. $\Box$

The previous theorem allows us to derive identities in a sumability algebra
$A=k+\mathfrak{m}$ with evaluations from universal identities in
$k\langle\\!\langle J\rangle\\!\rangle$ which only involve finite products and
infinite sums. We apply this in the next two results.

The next proposition extends this to identities involving composition of
formal power series (i.e. evaluations of formal power series at formal power
series). We will apply this to the exponential and logarithmic series in the
next subsection.

######  Proposition 2.7.

Let $A=k+\mathfrak{m}$ be a summability algebra that has evaluations. Let
$I,J$ be sets. Consider two families $\mathbf{Q}:I\longrightarrow
k\langle\\!\langle J\rangle\\!\rangle_{0}$ and
$\mathbf{f}:J\longrightarrow\mathfrak{m}$ which are summable in their
respective algebras. For $P\in k\langle\\!\langle I\rangle\\!\rangle$, we have

$\operatorname{ev}_{\mathbf{f}}(\operatorname{ev}_{\mathbf{Q}}(P))=\operatorname{ev}_{(\operatorname{ev}_{\mathbf{f}}(\mathbf{Q}(i)))_{i\in
I}}(P).$

Proof.  Let $i_{0}\in I$. We have
$\operatorname{ev}_{\mathbf{Q}}(X_{i_{0}})=\mathbf{Q}(i_{0})$, so

$\operatorname{ev}_{\mathbf{f}}(\operatorname{ev}_{\mathbf{Q}}(X_{i_{0}}))=\operatorname{ev}_{\mathbf{f}}(\mathbf{Q}(i_{0}))=\operatorname{ev}_{(\operatorname{ev}_{\mathbf{f}}(\mathbf{Q}(i)))_{i\in
I}}(X_{i_{0}}).$

Since $\operatorname{ev}_{\mathbf{f}}\circ\operatorname{ev}_{\mathbf{Q}}$ and
$\operatorname{ev}_{(\operatorname{ev}_{\mathbf{f}}(\mathbf{Q}(i)))_{i\in I}}$
are strongly linear morphisms of algebras, we deduce that the identity holds
for all $P\in k\langle\\!\langle I\rangle\\!\rangle$. $\Box$

######  Proposition 2.8.

Let $A=k+\mathfrak{m}$ have evaluations. Then $A$ is local algebra with
maximal ideal $\mathfrak{m}$.

Proof.  We first show that each element of $1+\mathfrak{m}$ is invertible in
$A$. Let $a=1+\varepsilon\in 1+\mathfrak{m}$. Since $(A,\Sigma)$ has
evaluations, writing

$P:=1-X_{0}+X_{0}^{2}-X_{0}^{3}+\cdots\in k\langle\\!\langle
1\rangle\\!\rangle,$

we may consider the evaluation $\operatorname{ev}_{\varepsilon}(P)\in A$. We
have $P(1+X_{0})=(1+X_{0})P=1$ in $k\langle\\!\langle 1\rangle\\!\rangle$, so
Theorem 2.6 gives
$\operatorname{ev}_{\varepsilon}(P)\cdot(1+\varepsilon)=(1+\varepsilon)\cdot\operatorname{ev}_{\varepsilon}(P)=1$,
i.e. $1+\varepsilon$ is a unit in $A$ with inverse
$\operatorname{ev}_{\varepsilon}(P)$.

We deduce that $k^{\times}(1+\mathfrak{m})=k^{\times}+\mathfrak{m}$ is
contained in $U(A)$. Since $\mathfrak{m}$ is a proper ideal, it follows that
$A\setminus U(A)=\mathfrak{m}$. So $A\setminus U(A)$ is an ideal, whence $A$
is local with maximal ideal $\mathfrak{m}$. $\Box$

###  2.3 Exponential and logarithm

In the sequel of Section 2, we assume that $k$ has characteristic zero. It is
known [25] that given a finite set $J$, the algebra $k\langle\\!\langle
J\rangle\\!\rangle$ is equipped with an exponential $\exp\colon
k\langle\\!\langle J\rangle\\!\rangle_{0}\longrightarrow 1+k\langle\\!\langle
J\rangle\\!\rangle_{0}$ and a logarithm $\log\colon 1+k\langle\\!\langle
J\rangle\\!\rangle_{0}\longrightarrow k\langle\\!\langle
J\rangle\\!\rangle_{0}$, which are inverses of one another, and are given by
evaluating the usual formal series. Using our previous results, we will
recover a number of known identities known in that case for all summability
algebras with evaluations.

We consider two particular elements of $k\langle\\!\langle 1\rangle\\!\rangle$
defined as follows

$E_{0}:=\sum_{n\geqslant 0}\frac{1}{n!}X_{0}^{n}\text{{\hskip
30.00005pt}and{\hskip 30.00005pt}$L_{0}:=\sum_{n\geqslant
1}\frac{(-1)^{n+1}}{n}X_{0}^{n}.$}$

Given a summability algebra $(A,\Sigma)$ which has evaluations, and writing
$\mathfrak{m}$ for its maximal ideal, we have two functions

$\displaystyle\exp:\mathfrak{m}$ $\displaystyle\longrightarrow$ $\displaystyle
1+\mathfrak{m}$ $\displaystyle\delta$ $\displaystyle\longmapsto$
$\displaystyle\operatorname{ev}_{\delta}(E_{0})$

and

$\displaystyle\log:1+\mathfrak{m}$ $\displaystyle\longrightarrow$
$\displaystyle\mathfrak{m}$ $\displaystyle\delta$ $\displaystyle\longmapsto$
$\displaystyle\operatorname{ev}_{\delta-1}(L_{0}).$

######  Proposition 2.9.

[25, Section 1.7, Theorem 7.2] We have

$\operatorname{ev}_{L_{0}}(E_{0}-1)=\operatorname{ev}_{E_{0}-1}(L_{0})=X_{0}.$

######  Corollary 2.10.

Let $(A,\Sigma)$ be a summability algebra with evaluations, with maximal ideal
$\mathfrak{m}$. Then $\exp:\mathfrak{m}\longrightarrow 1+\mathfrak{m}$ and
$\log:1+\mathfrak{m}\longrightarrow\mathfrak{m}$ are bijective, and are
functional inverses of one another.

Proof.  This follows from Proposition 2.7. For instance, for
$\varepsilon\in\mathfrak{m}$, we have

$\log(\exp(\varepsilon))=\operatorname{ev}_{\operatorname{ev}_{\varepsilon}(E_{0}-1)}(L_{0})=\operatorname{ev}_{\varepsilon}(\operatorname{ev}_{L_{0}}(E_{0}-1))=\operatorname{ev}_{\varepsilon}(X_{0})=\varepsilon.$

The other identity follows similarly. $\Box$

Next consider the following elements of $k\langle\\!\langle
2\rangle\\!\rangle$ for $n\in\mathbb{N}$:

$\displaystyle E_{1}$ $\displaystyle:=$
$\displaystyle\exp(X_{1})=\sum_{n\in\mathbb{N}}\frac{1}{n!}X_{1}^{n}$
$\displaystyle K_{n}$ $\displaystyle:=$
$\displaystyle\sum_{\begin{subarray}{c}m_{1}+p_{1},\cdots,m_{i}+p_{i}\geqslant
1\\\
m_{1}+p_{1}+\ldots+m_{i}+p_{i}=n\end{subarray}}\frac{1}{m_{1}!p_{1}!\cdots
m_{n}!p_{n}!}X_{0}^{m_{1}}X_{1}^{p_{1}}\cdots X_{0}^{m_{n}}X_{1}^{p_{n}}\ $
(6) $\displaystyle X_{0}\ast X_{1}$ $\displaystyle:=$
$\displaystyle\sum_{n\char 62\relax 0}\frac{(-1)^{n+1}}{n}K_{n}.$

We have the formal Baker–Campbell–Hausdorff Theorem:

######  Proposition 2.11.

[25, Section 1.8] We have $\operatorname{ev}_{E_{0}\cdot
E_{1}-1}(L_{0})=X_{0}\ast X_{1}$. Moreover, $K_{0}=X_{0}+X_{1}$ and each
$K_{n},n\char 62\relax 0$ lies in the Lie subalgebra of $(k\langle\\!\langle
2\rangle\\!\rangle,+,0,[\cdot,\cdot])$ generated by commutators of $X_{0}$ and
$X_{1}$.

Let $A=k+\mathfrak{m}$ have evaluations. For all $\delta_{1}$, $\delta_{2}$ in
$\mathfrak{m}$, we define

$\displaystyle\delta_{1}\ast\delta_{2}$ $\displaystyle:=$
$\displaystyle\operatorname{ev}_{\delta_{1},\delta_{2}}(X_{0}\ast
X_{1})\in\mathfrak{m}.$ (7)

As a consequence of Proposition 2.7, we have:

######  Corollary 2.12.

Let $(A,\Sigma)$ be a summability algebra with evaluations, and let
$\mathfrak{m}$ denote its maximal ideal. Then for all
$\delta_{1},\delta_{2}\in\mathfrak{m}$, we have
$\exp(\delta_{1})\cdot\exp(\delta_{2})=\exp(\delta_{1}\ast\delta_{2})$.

###  2.4 A group isomorphism between derivations and automorphisms

Let $(A,\Sigma)$ be an ultrafinite, unital summability algebra. Let
$\mathfrak{m}\subseteq\operatorname{Lin}^{+}(A)$ be a closed subalgebra such
that $k\operatorname{Id}_{A}+\ \mathfrak{m}$ has evaluations.

######  Proposition 2.13.

The exponential induces a bijection between
$\operatorname{Der}^{+}(A)\cap\mathfrak{m}$ and
$\operatorname{End}^{+}(A)\cap\operatorname{Id}_{A}+\mathfrak{m}$.

Proof.  Our proof is a slight adaptation of [21, Theorem 4] to our formal
context. Let $\partial\in\operatorname{Der}^{+}(A)\cap\mathfrak{m}$ and let
$a,b\in A$. For $n\in\mathbb{N}$, an easy induction using the Leibniz product
rule shows that

$\partial^{[n]}(a\cdot
b)=\sum_{i=0}^{n}\binom{n}{i}\partial^{[i]}(a)\cdot\partial^{[n-i]}(b).$

We have

$\displaystyle\exp(\partial)(a)\cdot\exp(\partial)(b)$
$\displaystyle=\left(\sum_{m\in\mathbb{N}}\frac{1}{m!}\partial^{[m]}(a)\right)\cdot\left(\sum_{p\in\mathbb{N}}\frac{1}{p!}\partial^{[p]}(b)\right)$
$\displaystyle=\sum_{m,p\in\mathbb{N}}\frac{1}{m!p!}\partial^{[m]}(a)\cdot\partial^{[p]}(b)$
(SA)
$\displaystyle=\sum_{n\in\mathbb{N}}\sum_{m+p=n}\frac{1}{m!p!}\partial^{[m]}(a)\cdot\partial^{[p]}(b)$
(SS3)
$\displaystyle=\sum_{n\in\mathbb{N}}\sum_{i=0}^{n}\frac{1}{i!(n-i)!}\partial^{[i]}(a)\cdot\partial^{[n-i]}(b)$
$\displaystyle=\sum_{n\in\mathbb{N}}\frac{\partial^{[n]}(a\cdot b)}{k!}$
$\displaystyle=\exp(\partial)(a\cdot b).$

So $\exp(\partial)\in\operatorname{End}^{+}(A)$. Conversely, let
$\sigma\in\operatorname{End}^{+}(A)\cap\operatorname{Id}_{A}+\mathfrak{m}$ and
write $\varepsilon:=\operatorname{Id}_{A}-\sigma\in\mathfrak{m}$. Let $a,b\in
A$. As in the proof of [21, Theorem 4], there is a family
$(c_{k,l,n})_{k,l,n\in\mathbb{N}}\in\mathbb{Q}^{\mathbb{N}^{3}}$ such that for
all $n\char 62\relax 0$, we have both

$\displaystyle\varepsilon^{[n]}(a\cdot b)$ $\displaystyle=$
$\displaystyle\sum_{l=0}^{n}\sum_{m=0}^{l}c_{n,l,m}\varepsilon^{[m]}(a)\cdot\varepsilon^{[l-m]}(b)\text{\qquad
in $A$, and}$ $\displaystyle(z_{1}+z_{2}-z_{1}z_{2})^{n}$ $\displaystyle=$
$\displaystyle\sum_{l=0}^{n}\sum_{m=0}^{l}c_{n,l,m}z_{1}^{m}z_{2}^{l-m}\text{\qquad
in $\mathbb{Q}[[z_{1},z_{2}]]$}.$

Note that given $l,m\in\mathbb{N}$ we have

$\forall n\char 62\relax l+m,c_{n,l,m}=0.$ (8)

So the sum $S_{l,m}:=\sum_{n=1}^{+\infty}\frac{1}{n}c_{n,l,m}$ has finite
support. We have

$\log(1-(z_{1}+z_{2}-z_{1}z_{2}))=\log(1-z_{1})+\log(1-z_{2})$

in $\mathbb{Q}[[z_{1},z_{2}]]$. Identifying in the left and right hand terms
the coefficients of $z_{1}^{p}z_{2}^{q}$ for $p\neq 0$ and $q\neq 0$, we
deduce ithat $S_{l,m}=0$ if $m\not\in\\{0,l\\}$ or $l=0$. Considering the
coefficients of $z_{1}^{l}$ and $z_{2}^{l}$ for $l\geqslant 1$, we see that
$S_{l,0}=S_{l,1}=\frac{1}{l}$ otherwise. Now

$\displaystyle\log(\sigma)(a\cdot b)$ $\displaystyle=\sum_{n\geqslant
1}\frac{1}{n}\sum_{l=0}^{n}\sum_{m=0}^{l}c_{n,l,m}\varepsilon^{[m]}(a)\cdot\varepsilon^{[l-m]}(b)$
$\displaystyle=\sum_{n\geqslant 1}\sum_{m\leqslant l\leqslant
n}^{n}\frac{c_{n,l,m}}{n}\varepsilon^{[m]}(a)\cdot\varepsilon^{[l-m]}(b)$
$\displaystyle=\sum_{n\geqslant 1\wedge m\leqslant l\leqslant
n}\frac{c_{n,l,m}}{n}\varepsilon^{[m]}(a)\cdot\varepsilon^{[l-m]}(b)$
(Proposition 1.6) $\displaystyle=\sum_{l\geqslant 0\wedge m\leqslant l\wedge
n\geqslant l\wedge n\geqslant
1}\frac{c_{n,l,m}}{n}\varepsilon^{[m]}(a)\cdot\varepsilon^{[l-m]}(b)$ (SS2)
$\displaystyle=\sum_{l\geqslant 0}\sum_{n\geqslant 1\wedge n\geqslant
l\geqslant
m}\frac{c_{n,l,m}}{n}\varepsilon^{[m]}(a)\cdot\varepsilon^{[l-m]}(b)$
(Proposition 1.6) $\displaystyle=\sum_{l\geqslant 0}\sum_{m\leqslant
l}\sum_{n\geqslant 1\wedge n\geqslant l\wedge n\leqslant
l+m}\frac{c_{n,l,m}}{n}\varepsilon^{[m]}(a)\cdot\varepsilon^{[l-m]}(b)$ ((8))
$\displaystyle=\sum_{l\geqslant 0}\sum_{m\leqslant
l}S_{l,m}\varepsilon^{[m]}(a)\cdot\varepsilon^{[l-m]}(b)$
$\displaystyle=\sum_{l\geqslant
1}\frac{1}{l}\sum_{m\in\\{0,l\\}}\varepsilon^{[m]}(a)\cdot\varepsilon^{[l-m]}(b)$
$\displaystyle=\sum_{l\geqslant
1}\frac{1}{l}(a\cdot\varepsilon^{[l]}(b)+\varepsilon^{[l]}(a)\cdot b)$
$\displaystyle=a\cdot\log(1-\varepsilon)(b)+\log(1-\varepsilon)(a)\cdot b$
$\displaystyle=a\cdot\log(\sigma)(b)+\log(\sigma)(a)\cdot b.$

Therefore $\log(\sigma)$ is a derivation. $\Box$

######  Theorem 2.14.

Consider the operation $\ast$ of eq. 7 on $\mathfrak{m}$. The structures
$(\operatorname{Der}^{+}(A)\cap\mathfrak{m},\ast)$ and
$(\operatorname{End}^{+}(A)\cap\operatorname{Id}_{A}+\mathfrak{m},\circ)$ are
groups, and the exponential map of Proposition 2.13 is a group isomorphism.

Proof.  By Proposition 2.8, each $\sigma\in\operatorname{Id}_{A}+\mathfrak{m}$
is invertible in $\operatorname{Id}_{A}+\mathfrak{m}$, and its inverse is
obviously a morphism of algebra of $A$, so
$\operatorname{End}^{+}(A)\cap\operatorname{Id}_{A}+\mathfrak{m}$ is a group
under composition. We conclude with Corollary 2.12 and Proposition 2.13.
$\Box$

######  Corollary 2.15.

The group
$(\operatorname{End}^{+}(A)\cap\operatorname{Id}_{A}+\mathfrak{m},\circ)$ is
divisible and torsion-free.

Proof.  Let $n\in\mathbb{N}^{\char 62\relax 0}$ and
$a\in\operatorname{Der}^{+}(A)\cap\mathfrak{m}$. Note that the $n$-fold
iterate of $a$ in $(\operatorname{Der}^{+}(A),\ast)$ is $a\ast a\cdot\ast
a=na$. It follows since $k$ has haracteristic zero that
$(\operatorname{Der}^{+}(A)\cap\mathfrak{m},\ast)$ is torsion-free.
Furthermore, we see that the $n$-fold iterate of $\frac{1}{n}a$ in
$(\operatorname{Der}^{+}(A),\ast)$ is $a$, whence
$(\operatorname{Der}^{+}(A)\cap\mathfrak{m},\ast)$ is divisible. We conclude
with Theorem 2.14. $\Box$

######  Proposition 2.16.

For all $\sigma\in\operatorname{Id}_{A}+\mathfrak{m}$, writing
$\mathcal{C}(\sigma)=\\{\mu\in\operatorname{End}^{+}(A)\cap\operatorname{Id}_{A}+\mathfrak{m}:\mu\circ\sigma=\sigma\circ\mu\\}$,
we have a group morphism

$\displaystyle\mathord{\>{}^{[\cdot]}}:(k,+,0)$ $\displaystyle\longrightarrow$
$\displaystyle(\mathcal{C}(\sigma),\circ,\operatorname{Id}_{A})$
$\displaystyle c$ $\displaystyle\longmapsto$
$\displaystyle\sigma^{[c]}:=\exp(c\log(\sigma)),$

with $\sigma^{[1]}=\sigma$ and
$(\sigma^{[c]})^{[c^{\prime}]}=\sigma^{[cc^{\prime}]}$ for all
$c,c^{\prime}\in k$. It is injective if $\sigma\neq\operatorname{Id}_{A}$.

Proof.  Write
$\partial:=\log(\sigma)\in\operatorname{Der}_{\prec}^{+}(\mathbb{A})$ and let
$c\in k$. Recall by Proposition 2.11 that for $n\in\mathbb{N}^{\char 62\relax
0}$, the terms $\operatorname{ev}_{(\partial,c\partial)}(K_{n})$ and
$\operatorname{ev}_{(c\partial,\partial)}(K_{n})$, where $K_{n}$ is as in (6),
lie in the Lie algebra generated by commutators in $\partial$ and $c\partial$.
All such commutators are zero, so
$\operatorname{ev}_{(\partial,c\partial)}(K_{n})=\operatorname{ev}_{(c\partial,\partial)}(K_{n})=0$.
It follows since
$\operatorname{ev}_{(\partial,c\partial)}(K_{0})=\partial+c\partial=\operatorname{ev}_{(c\partial,\partial)}(K_{0})$
that $(c\partial)\ast\partial=(c+1)\partial=\partial\ast(c\partial)$, so
$c\partial$ commutes with $\partial$, whence
$\exp(c\partial)=\sigma^{[c]}\in\mathcal{C}(\sigma)$.

For $c,c^{\prime}\in k$, we have
$\sigma^{[c+c^{\prime}]}=\exp(c\partial+c^{\prime}\partial)=\exp((c\partial)\ast(c^{\prime}\partial))$
as above. So
$\sigma^{[c+c^{\prime}]}=\exp(c\partial)\circ\exp(c^{\prime}\partial)=\sigma^{[c]}\circ\sigma^{[c^{\prime}]}$.
Thus $\mathord{\>{}^{[\cdot]}}$ is a group morphism. We also have
$\sigma^{[cc^{\prime}]}=\exp(c^{\prime}\log(\exp(c\log(\sigma))))=\exp(cc^{\prime}\log(\sigma))=\sigma^{[cc^{\prime}]}$.

Assume that $\sigma\neq\operatorname{Id}_{A}$, so $\log(\sigma)\neq 0$. The
kernel of the morphism is

$\\{c\in k:\exp(c\log(\sigma))=\operatorname{Id}_{A}\\}=\\{c\in
k:c\log(\sigma)=0\\}=\\{0\\}.$

So this morphism is injective. $\Box$

##  3 Application to Noetherian series

###  3.1 Noetherian orderings

######  Definition 3.1.

Let $(\Omega,<)$ be a partially ordered set. A chain in $(\Omega,<)$ is a
subset of $\Omega$ which is linearly ordered by the induced ordering. A
decreasing chain in $(\Omega,<)$ is chain $Y\subseteq\Omega$ without minimal
element, i.e. with

$\forall y\in Y,\exists z\in Y,(z<y).$

An antichain in $(\Omega,<)$ is a subset $Y\subseteq\Omega$, no two distinct
elements of which are comparable, i.e. with

$\forall y,z\in Y,(y<z\vee y=z)\Longrightarrow y=z.$

We say that $(\Omega,<)$ is Noetherian, or that $<$ is a Noetherian ordering
on $\Omega$, if there are no infinite decreasing chains and no infinite
antichains in $(\Omega,<)$.

Note that linear Noetherian orderings are exactly well-orderings.

######  Proposition 3.2.

[11, Proposition A.1] An ordered set $(\Omega,<)$ is Noetherian if and only if
every sequence $u:\mathbb{N}\longrightarrow\Omega$ has an increasing
subsequence.

If $(X,<_{X})$ is a partially ordered set, then a bad sequence in $X$ is a
sequence $u:\mathbb{N}\longrightarrow X$ such that there are no numbers
$i,j\in\mathbb{N}$ with $i<j$ and $u_{i}\leqslant_{X}u_{j}$. Given a function
$f:X\longrightarrow\mathbb{N}$, a bad sequence $u$ in $\mathbf{X}$ is said
minimal for $f$ if for all $i\in\mathbb{N}$, there are no bad sequences $v$ in
$X$ with $(v_{0},\ldots,v_{i-1})=(u_{0},\ldots,u_{i-1})$ and
$f(v_{i})<f(u_{i})$. If there is a bad sequence in $X$, then there is a
minimal one for $f$ (see [11, p 307]).

######  Lemma 3.3.

[10, Theorem 2.1] Let $(X,<)$ be a partially ordered set. The following
statements are equivalent

  1. a)

$(X,<)$ is Noetherian.

  2. b)

There is no bad sequence in $(X,<)$.

See [10, Theorem 2.1] for other characterizations of Noetherian orderings.

######  Lemma 3.4.

[10, Theorem 2.3] Let $(\Omega_{1},<_{1})$ and $(\Omega_{2},<_{2})$ be
Noetherian ordered sets. Then their product $\Omega_{1}\times\Omega_{2}$ is
Noetherian for the ordering

$(p_{1},p_{2})<(q_{1},q_{2})\Longleftrightarrow((p_{1},p_{2})\neq(q_{1},q_{2})\wedge
p_{1}\leqslant p_{2}\wedge q_{1}\leqslant q_{2}).$

######  Proposition 3.5.

Let $(\Omega,<)$ be an ordered set. Then the set $\mathcal{N}$ of Noetherian
subsets of $\Omega$ is a bornology on $\Omega$.

Proof.  If $X\subseteq Y$ are subsets of $\Omega$, then a decreasing chain
(resp. an antichain) in $X$ is a decreasing chain (resp. an antichain) in $Y$.
So $X$ is Noetherian if $Y$ is Noetherian. Let $X_{1},X_{2}$ be Noetherian
subsets of $\Omega$. If $C$ were an infinite decreasing chain in $X_{1}\cup
X_{2}$, then $C\cap X_{1}$ or $C\cap X_{2}$ would be a decreasing chain in
$X_{1}$ or $X_{2}$ respectively, which cannot be. If $A$ is an antichain in
$X_{1}\cup X_{2}$, then $A\cap X_{1}$ and $A\cap X_{2}$ are antichains in
$X_{1}$ and $X_{2}$ respectively, so $A$ must be finite. So $X_{1}\cup X_{2}$
is Noetherian. Thus $\mathcal{N}$ is a bornology on $\Omega$. $\Box$

We next state a weaker and simplified version of van der Hoeven’s theorem [11,
Appendix A.4]. A function $\vartheta$ sending each $p\in\Omega$ to a subset
$\vartheta(p)$ of $\Omega$ is called a choice operator on $\Omega$. The choice
operator $\vartheta$ is said Noetherian if for all Noetherian subsets
$Y\subseteq\Omega$, the set

$Y_{\vartheta}:=\bigcup_{y\in Y}\vartheta(y)\subseteq\Omega$

is Noetherian. It is said strictly extensive if for all $p\in\Omega$, we have

$p<\vartheta(p)\text{, \quad that is,\quad$\forall y\in\vartheta(p),p<y.$}$

For any non-empty word
$w=(w_{0},\ldots,w_{m})\in\Omega^{\star}\setminus\\{\emptyset\\}$, we write
$w_{\bullet}:=w_{m}\in\Omega$ for the last “letter” of $w$. Let
$Y\subseteq\Omega$ be a subset. Let $\vartheta^{+}(Y)$ denote the set of non-
empty words $(w_{0},\ldots,w_{n})\in\Omega^{\star}\setminus\\{\emptyset\\}$
where $w_{0}\in Y$, and for each $i<n$, we have $w_{i+1}\in\vartheta(w_{i})$.
We endow $\vartheta^{+}(Y)$ with the ordering $<_{\vartheta}$ defined by

$w<_{\vartheta}w^{\prime}\Longleftrightarrow
w_{\bullet}<w^{\prime}_{\bullet}.$

######  Proposition 3.6.

Let $\vartheta$ be a Noetherian and strictly extensive choice operator on
$\Omega$. Then for all Noetherian subsets $Y$ of $\Omega$, the set
$\vartheta^{+}(Y)$ is Noetherian for $<_{\vartheta}$.

Proof.  This version follows from an application of [11, Theorem A.4] to a
simple case. Nonetheless, let us adapt van der Hoeven’s proof to the present
simplified setting.

Assume for contradiction that $\vartheta^{+}(Y)$ is not Noetherian. So there
is a minimal bad sequence
$(w_{i})_{i\in\mathbb{N}}\in\vartheta^{+}(Y)^{\mathbb{N}}$ for the length
function $w\mapsto|w|$. Assume that there is an infinite set
$I\subseteq\mathbb{N}$ with $|w_{i}|=2$ for all $i\in I$. Then
$\mathcal{Y}:=\\{w_{i,0}:i\in I\\}\subseteq Y$ is Noetherian. Since
$\vartheta$ is a Noetherian choice operator, the set

$\mathcal{Y}_{\vartheta}:=\bigcup_{i\in I}\vartheta(x_{i,0})$

is Noetherian. But then $\\{w_{i}:i\in I\\}$ is Noetherian for
$<_{\vartheta}$: a contradiction.

So there is an $m\in\mathbb{N}$ with $|w_{j}|\char 62\relax 2$ for all
$j\geqslant m$. For $j\geqslant m$, we write
$z_{j}:=(w_{j,0},\ldots,w_{j,|w_{j}|-2})\in\vartheta^{+}(Y)$. We claim that
the set $\mathcal{Z}:=\\{z_{j}:j\geqslant m\\}$, is Noetherian for
$<_{\vartheta}$. Indeed, assume for contradiction that
$(z_{j_{i}})_{i\in\mathbb{N}}$ is a bad sequence in $\mathcal{Z}$ with
$j_{0}\leqslant j_{1}\leqslant\cdots$. We further claim that

$u:=(w_{0},\ldots,w_{j_{0}-1},z_{j_{0}},z_{j_{1}},\ldots)$

is a bad sequence, contradicting the minimality of $(w_{i})_{i\in\mathbb{N}}$.
Assume for contradiction that $u$ is not bad. Since
$(z_{j_{i}})_{i\in\mathbb{N}}$ is bad, there must exist $i<j_{0}$ and
$p\in\mathbb{N}$ with $w_{i}\leqslant_{\vartheta}z_{j_{p}}$. Since $\vartheta$
is strictly extensive, we have

$(w_{j_{p}})_{\bullet}\in\vartheta((z_{j_{p}})_{\bullet})\char
62\relax(z_{j_{p}})_{\bullet},$

so $w_{i}<_{\vartheta}w_{m}$: a contradiction. Therefore $\mathcal{Z}$ is
Noetherian. It follows since $\vartheta$ is Noetherian that
$\\{w_{i}:i\geqslant m\\}$ is Noetherian: a contradiction. $\Box$

###  3.2 Spaces of Noetherian series

We fix an ordered set $(\Omega,<)$ and a summability space $V$ with the
minimal summability structure. We have an ultrafinite summability space
$(V^{[\mathcal{N}]},\Sigma^{\mathcal{N}})$ where $\mathcal{N}$ is the
bornology of Noetherian subsets of $\Omega$. We say that
$(V^{[\mathcal{N}]},\Sigma^{\mathcal{N}})$ is a space of Noetherian series.

######  Lemma 3.7.

Let $I$ be a set and let $\mathbf{f}:I\longrightarrow V^{[\mathcal{N}]}$ be a
function. Consider the set

$N_{\mathbf{f}}:=\\{(i,p)\in
I\times\Omega:p\in\operatorname{supp}\mathbf{f}(i)\\},$

ordered by $(i,p)<_{\mathbf{f}}(j,q)\Longleftrightarrow p<q$. Then
$\mathbf{f}$ is $\mathcal{N}$-summable if and only if
$(N_{\mathbf{f}},<_{\mathbf{f}})$ is Noetherian.

Proof.  Consider a non-empty chain $C$ for $(N_{\mathbf{f}},<_{\mathbf{f}})$.
Given $(i,p)\in C$, we have $p\in\bigcup_{i\in
I}\operatorname{supp}\mathbf{f}(i)$, and $(i,p)$ is $<_{\mathbf{f}}$ minimal
in $C$ if and only if $p$ is minimal in $\bigcup_{i\in
I}\operatorname{supp}\mathbf{f}(i)$. So $N_{\mathbf{f}}$ as infinite
decreasing chains if and only if $\bigcup_{i\in
I}\operatorname{supp}\mathbf{f}(i)$ has infinite decreasing chains.

Consider an antichain $A$ in $(N_{\mathbf{f}},<_{\mathbf{f}})$. For
$(i,p),(j,q)\in A$, either $p=q$ and $i\neq j$ or $p$ and $q$ are not
comparable in $(\Omega,<)$. So $N_{\mathbf{f}}$ has an infinite antichain if
and only if there is an infinite antichain in $\bigcup_{i\in
I}\operatorname{supp}\mathbf{f}(i)$ or there is an $p\in\bigcup_{i\in
I}\operatorname{supp}\mathbf{f}(i)$ such that the set $I_{p}=\\{i\in
I:p\in\operatorname{supp}\mathbf{f}(i)\\}$ is infinite.

In view of the definitions of $\mathcal{N}$-summability and Noetherian
orderings, we deduce that $\mathbf{f}$ is $\mathcal{N}$-summable if and only
if $(N_{\mathbf{f}},<_{\mathbf{f}})$ is Noetherian. $\Box$

###  3.3 Algebras of Noetherian series

We fix an ordered monoid $(M,+,0,<)$, i.e. a monoid $(M,+,0)$ together with an
ordering $<$ on $M$ with

$\forall f,g,h\in M,f<g\Longrightarrow(f+h<g+h\wedge h+f<h+g).$ (9)

Let $\mathcal{N}$ denote the bornology of Noetherian subsets of $(M,<)$. We
write $k(\\!(M)\\!):=k^{[\mathcal{N}]}$, with its pointwise summmability
structure $\Sigma^{\mathcal{N}}$ with respect to the minimal summmability
structure on $k$. As in [24, 19], given $g\in M$, we write $t^{g}$ for the
function $M\longrightarrow k$ with support $\\{g\\}$ and value $1$ at $g$,
i.e. $t^{g}=\mathbbm{1}_{\\{g\\}}$. Recall by (2) that for each $a\in
k(\\!(M)\\!)$, the family $(a(g)t^{g})_{g\in M}$ is $\mathcal{N}$-summable,
with sum

$a=\sum_{g\in M}a(g)t^{g}.$

The vector space $k(\\!(M)\\!)$ is equipped with the Cauchy product

$\forall g\in M,(a\cdot b)(g):=\sum_{f+h=g}a(g_{1})b(g_{2}).$ (10)

######  Lemma 3.8.

The function $a\cdot b$ is well-defined and lies in $k(\\!(M)\\!)$.

Proof.  We first show that for $g\in M$, the set

$I:=\\{(f,h)\in(\operatorname{supp}a)\times(\operatorname{supp}b):f+h=g\\}$

is finite. Assume for contradiction that it is infinite. If its projection
$I_{1}$ on the first variable is infinite, then we find an injective sequence
$f:\mathbb{N}\longrightarrow I_{1}$. Let
$h:\mathbb{N}\longrightarrow\operatorname{supp}b$ be a sequence with
$(f(n),h(n))\in I$ for all $n\in\mathbb{N}$. Since
$I_{1}\subseteq\operatorname{supp}a$ is Noetherian, by Proposition 3.2, there
is an increasing subsequence $f\circ\varphi$ of $f$. Likewise, there is an
increasing subsequence $h\circ\varphi\circ\psi$ of $h\circ\varphi$. We have
$g=f\circ\varphi\circ\psi(1)+h\circ\varphi\circ\psi(1)\char 62\relax
f\circ\varphi\circ\psi(0)+h\circ\varphi\circ\psi(1)$ and
$f\circ\varphi\circ\psi(0)+h\circ\varphi\circ\psi(1)\geqslant
f\circ\varphi\circ\psi(0)+h\circ\varphi\circ\psi(0)=g$ by (9), so $g\char
62\relax g$: a contradiction.

If $I_{1}$ is finite, then the projection of $I$ on the second variable must
be infinite, and we obtain a symmetric contradiction. Therefore $I$ is finite.
So $a\cdot b$ is well-defined.

We next show that the set

$S:=\\{g\in
M:\exists(g_{1},g_{2})\in(\operatorname{supp}a)\times(\operatorname{supp}b),g=g_{1}+g_{2}\\}$

is Noetherian. It will then follow since $\operatorname{supp}(a\cdot
b)\subseteq S$ that $a\cdot b\in k(\\!(M)\\!)$. Let $(g_{n})_{n\in\mathbb{N}}$
be a sequence in $S$ and let $f\in(\operatorname{supp}a)^{\mathbb{N}}$ and
$g\in(\operatorname{supp}b)^{\mathbb{N}}$ with $g_{n}=f(n)+h(n)$ for all
$n\in\mathbb{N}$. By Proposition 3.2, there is an increasing subsequence
$f\circ\varphi$ of $f$, and an increasing subsequence $h\circ\varphi\circ\psi$
of $h\circ\varphi$. Thus
$(g_{\varphi(\psi(n))})_{n\in\mathbb{N}}=f\circ\varphi\circ\psi+h\circ\varphi\circ\psi$
is an increasing subsequence of $(g_{n})_{n\in\mathbb{N}}$. We deduce with
Proposition 3.2 that $S$ is Noetherian. $\Box$

######  Proposition 3.9.

For the Cauchy product, the structure $k(\\!(M)\\!)$ is a unital algebra with
multiplicative identity $1:=t^{0}$.

Proof.  Let $a,b,c\in k(\\!(M)\\!)$ and $g\in M$. We have

$\displaystyle(a\cdot b)\cdot c$ $\displaystyle=\sum_{g_{1}+g_{2}=g}(a\cdot
b)(g_{1})c(g_{2})$
$\displaystyle=\sum_{g_{1}+g_{2}=g}\left(\sum_{g_{3}+g_{4}=g_{1}}a(g_{3})b(g_{4})\right)c(g_{2})$
$\displaystyle=\sum_{g_{1}+g_{3}+g_{4}=g}a(g_{3})b(g_{4})c(g_{2}).$ ($(M,+)$
is associative)

Likewise $a\cdot(b\cdot
c)=\sum_{g_{1}+g_{3}+g_{4}=g}a(g_{3})b(g_{4})c(g_{2})=(a\cdot b)\cdot c$. So
the product is associative. It is clearly bilinear. It is easy to see that

$(t^{0}\cdot a)(g)=(a\cdot t^{0})(g)=a(g),$

so $1$ is the multiplicative identity in $k(\\!(M)\\!)$. $\Box$

######  Proposition 3.10.

The algebra $k(\\!(M)\\!)$ with its structure of bornological space is a
summability algebra.

Proof.  We have to show that SA holds. This follows as in [12, Proposition
3.3], where the commutativity of the monoid does not play a role. $\Box$

In particular, for $a,b\in k(\\!(M)\\!)$, the family
$(a(g_{1})b(g_{2})t^{g_{1}+g_{2}})_{g_{1},g_{2}\in M}$ is
$\mathcal{N}$-summable, with sum

$a\cdot b=\sum_{g_{1},g_{2}\in M}a(g_{1})b(g_{2})t^{g_{1}+g_{2}}.$

We call $k(\\!(M)\\!)$ the algebra of Noetherian series (with coefficients in
$k$ and exponents in $M$).

###  3.4 Contracting linear maps

In this subsection and the next one, we fix an ordered set $\Omega$, we
consider again the bornology $\mathcal{N}$ of Noetherian subsets of $\Omega$,
and we write $\mathbb{V}:=k^{[\mathcal{N}]}$ for the corresponding
bornological space. Given $v,w\in\mathbb{V}$, we write

$v\prec w$

if $w\neq 0$ and for all $p\in\operatorname{supp}v$, there is a
$q\in\operatorname{supp}w$ with $p\char 62\relax q$.

######  Proposition 3.11.

[12, Proposition 4.6] The relation $\prec$ is an ordering on $\mathbb{V}$.
Moreover, for $u,v,w\in\mathbb{V}$, we have $u\prec w\wedge v\prec
w\Longrightarrow u+v\prec w$.

######  Definition 3.12.

A function $\phi:\mathbb{V}\longrightarrow\mathbb{V}$ is said contracting if
for all $a,b\in\mathbb{V}$, we have

$v\neq w\Longrightarrow\phi(w)-\phi(v)\prec w-v.$

We write $\operatorname{Lin}_{\prec}(\mathbb{V})$ (resp.
$\operatorname{Lin}_{\prec}^{+}(\mathbb{V})$) for the sets of contracting
linear (resp. strongly linear) maps $\mathbb{V}\longrightarrow\mathbb{V}$.
Note that

$\operatorname{Lin}_{\prec}^{+}(\mathbb{V})=\operatorname{Lin}_{\prec}(\mathbb{V})\cap\operatorname{Lin}^{+}(\mathbb{V}).$
(11)

######  Lemma 3.13.

$\operatorname{Lin}_{\prec}(\mathbb{V})$ is a subalgebra of
$\operatorname{Lin}(\mathbb{V})$.

Proof.  If $\phi,\psi\in\operatorname{Lin}_{\prec}(\mathbb{V})$ and $c\in k$,
then for $v\in\mathbb{V}\setminus\\{0\\}$, we have
$\operatorname{supp}c\psi(v)\subseteq\operatorname{supp}\psi(v)$ so
$c\psi(v)\prec v$. We deduce with Proposition 3.11 that $\phi(v)+c\psi(v)\prec
v$, whence $\phi+c\psi\in\operatorname{Lin}_{\prec}(\mathbb{V})$. If
$\psi(v)=0$, then $\phi(\psi(v))=0\prec v$. Otherwise
$\phi(\psi(v))\prec\psi(v)$ where $\psi(v)\prec v$ so $\phi\circ\psi(v)\prec
v$ by Proposition 3.11, whence
$\phi\circ\psi\in\operatorname{Lin}_{\prec}(\mathbb{V})$. $\Box$

######  Lemma 3.14.

If $\phi:\mathbb{V}\longrightarrow\mathbb{V}$ is strongly linear, then it is
contracting if and only if
$\operatorname{supp}\phi(\mathbbm{1}_{\\{p\\}})\char 62\relax p$ for all
$p\in\Omega$.

Proof.  Assume that $\phi$ satisfies the condition above, and let
$v\in\mathbb{V}$ be non-zero. By strong linearity, for each
$p\in\operatorname{supp}\phi(v)$, there is a $q\in\operatorname{supp}v$ with
$p\in\operatorname{supp}\phi(\mathbbm{1}_{\\{q\\}})$. Thus there is an
$r\in\operatorname{supp}\mathbbm{1}_{\\{q\\}}$ with $p\char 62\relax u$. But
$\operatorname{supp}\mathbbm{1}_{\\{q\\}}=\\{q\\}$ so $r=q$, whence $p\char
62\relax q\in\operatorname{supp}v$. This shows that $\phi(v)\prec v$. Since
$\phi$ is linear, it follows that $\phi$ is contracting. The converse is
immediate since $\phi(\mathbbm{1}_{\\{p\\}})\prec\mathbbm{1}_{\\{p\\}}$ is
equivalent to $\operatorname{supp}\phi(\mathbbm{1}_{\\{p\\}})\char 62\relax
p$. $\Box$

######  Corollary 3.15.

$\operatorname{Lin}_{\prec}^{+}(\mathbb{V})$ is a closed subalgebra of
$\operatorname{Lin}^{+}(\mathbb{V})$ and a closed ideal of
$k\operatorname{Id}_{\mathbb{V}}+\operatorname{Lin}_{\prec}^{+}(\mathbb{V})$.

Proof.  That $\operatorname{Lin}_{\prec}^{+}(\mathbb{V})$ is a subalgebra
follows from Proposition 1.21, (11) and Lemma 3.13. Let
$\boldsymbol{\phi}\colon
J\longrightarrow\operatorname{Lin}_{\prec}^{+}(\mathbb{V})$ be
$\operatorname{Lin}$-summable and set

$\boldsymbol{\sigma}:=\sum_{j\in
J}\boldsymbol{\phi}(j)\in\operatorname{Lin}^{+}(\mathbb{V}).$

Let $p\in\Omega$. We have
$\operatorname{supp}\boldsymbol{\sigma}(\mathbbm{1}_{\\{p\\}})\subseteq\bigcup_{j\in
J}\operatorname{supp}\boldsymbol{\phi}(j)(\mathbbm{1}_{\\{p\\}})$. Lemma 3.14
gives $\operatorname{supp}\boldsymbol{\phi}(j)(\mathbbm{1}_{\\{p\\}})\char
62\relax p$ for each $j\in J$, so
$\operatorname{supp}\boldsymbol{\sigma}(\mathbbm{1}_{\\{p\\}})\char 62\relax
p$, so $\boldsymbol{\sigma}$ is contracting, again by Lemma 3.14. $\Box$

######  Theorem 3.16.

The algebra
$k\operatorname{Id}_{\mathbb{V}}+\operatorname{Lin}_{\prec}^{+}(\mathbb{V})$
has evaluations.

Proof.  Let $J$ be a set and let $(\phi_{j})_{j\in
J}\subseteq(\operatorname{Lin}_{\prec}^{+}(\mathbb{V}))^{J}$ be
$\operatorname{Lin}$-summable. Let us show that the family

$(\phi_{\theta_{1}}\circ\cdots\circ\phi_{\theta_{n}})_{(\theta_{1},\ldots,\theta_{n})\in
J^{\star}}$

is $\operatorname{Lin}$-summable. We may assume that $J$ is non-empty, and it
suffices to show that

$(\phi_{\theta_{0}}\circ\cdots\circ\phi_{\theta_{n}})_{(\theta_{0},\ldots,\theta_{n})\in
J^{\star}}$

is $\operatorname{Lin}$-summable.

Let $\pi:\Omega\times J\longrightarrow\Omega$ denote the projection on the
first variable. We consider the ordering $<$ on $\Omega\times J$ given by
$(p,i)<(q,j)\Longleftrightarrow p<q$. Consider the choice operator $\vartheta$
on $\Omega\times J$ given by

$\forall p\in\Omega,\vartheta(p,i):=\\{(q,j):j\in J\wedge
q\in\operatorname{supp}\phi_{j}(\mathbbm{1}_{\\{p\\}})\\}.$

Since each $\phi_{j}$ for $j\in J$ is contracting, this is a strictly
extensive choice operator. Let $Y\subseteq\Omega\times J$ be Noetherian. Let
us show that the set

$Y_{\vartheta}=\\{(q,j):j\in J\wedge(\exists
p\in\pi(Y),(q\in\operatorname{supp}\phi_{j}(\mathbbm{1}_{\\{p\\}})))\\}$

is Noetherian by showing that each sequence in $Y_{\vartheta}$ has an
increasing subsequence. Let $(q_{n},j_{n})_{n\in\mathbb{N}}$ be a sequence in
$Y_{\vartheta}$, and pick for each $n\in\mathbb{N}$ a
$(p_{n},j_{n}^{\prime})\in Y$ with
$q_{n}\in\operatorname{supp}\phi_{j_{n}}(\mathbbm{1}_{\\{p_{n}\\}})$. Since
$Y$ is Noetherian, we may assume by Proposition 3.2 that
$(p_{n},j_{n}^{\prime})_{n\in\mathbb{N}}$ is increasing.

Assume that $(p_{n})_{n\in\mathbb{N}}$ has a constant subsequence. Without
loss of generality, we may assume that it is constant itself. Assume that
$(j_{n})_{n\in\mathbb{N}}$ has no injective subsequence. So it has a constant
subsequence $(j_{\psi(n)})_{n\in\mathbb{N}}$. The sequence
$(q_{\psi(n)})_{n\in\mathbb{N}}$ in the Noetherian set
$\operatorname{supp}\phi_{j_{\psi(0)}}(\mathbbm{1}_{\\{p_{\psi(0)}\\}})$ has
an increasing subsequence $(q_{\psi\circ\mu(n)})_{n\in\mathbb{N}}$, and
$(q_{\psi\circ\mu(n)},j_{\psi(0)})_{n\in\mathbb{N}}$ is an increasing
subsequence of $(q_{n},j_{n})_{n\in\mathbb{N}}$. Assume now that
$(j_{n})_{n\in\mathbb{N}}$ has an injective subsequence. Since
$(\phi_{j_{n}})_{n\in\mathbb{N}}$ is $\operatorname{Lin}$-summable, the family
$(\phi_{j_{n}}(\mathbbm{1}_{\\{p_{0}\\}}))_{n\in\mathbb{N}}$ is
$\mathcal{N}$-summable. Therefore $(q_{n})_{n\in\mathbb{N}}$ must have a
strictly increasing subsequence, whence $(q_{n},j_{n})_{n\in\mathbb{N}}$ has a
strictly increasing subsequence.

Assume now that $(p_{n})_{n\in\mathbb{N}}$ has no constant subsequence. So
$(p_{n})_{n\in\mathbb{N}}$ has a strictly increasing subsequence
$(p_{\varphi(n)})_{n\in\mathbb{N}}$. The family
$(\phi_{j_{\varphi(n)}}(\mathbbm{1}_{\\{p_{\varphi(n)}\\}}))_{n\in\mathbb{N}}$
is $\mathcal{N}$-summable. In particular $(q_{\varphi(n)})_{n\in\mathbb{N}}$
has a strictly increasing subsequence, whence again
$(q_{n},j_{n})_{n\in\mathbb{N}}$ has a strictly increasing subsequence.

This shows that $\vartheta$ is Noetherian. Let $S\subseteq\Omega$ be a
Noetherian subset and pick an arbitrary $j\in J$. By Proposition 3.6, the set
$\vartheta^{+}(S\times\\{j\\})$ is Noetherian for $<_{\vartheta}$. This means
by Lemma 3.7 that the family
$(p_{\bullet})_{p\in\vartheta^{+}(S\times\\{j\\})}$ is $\mathcal{N}$-summable.
Write

$I:=\\{(p,n,\theta_{0},\ldots,\theta_{n}):p\in S\wedge
n\in\mathbb{N}\wedge\theta_{0},\ldots,\theta_{n}\in J\\}.$

For $i=(p,n,\theta_{0},\ldots,\theta_{n})\in I$, let $W_{i}$ denote the set of
words

$w=(w_{0},\ldots,w_{n+1})\in\vartheta^{+}(S\times\\{j\\})$

where $w_{0}=(p,j)$ and $w_{k+1}=(p_{k+1},\phi_{\theta_{k}})$ for a
$p_{k+1}\in\operatorname{supp}\phi_{\theta_{k}}(\mathbbm{1}_{\\{\pi(w_{k})\\}})$.
Note that for each $i=(p,n,\phi_{0},\ldots,\phi_{n})\in I$, there is a family
$c\in k^{W_{i}}$ such that $\phi_{n}\circ\cdots\circ\phi_{0}(p)=\sum_{w\in
W_{i}}c(w)\mathbbm{1}_{\\{\pi(w_{\bullet})\\}}$.

The sets $W_{i},i\in I$ are pairwise disjoint, so by SS4, Proposition 1.6 and
UF, the family $\left(\sum_{w\in
W_{i}}c(w)\mathbbm{1}_{\\{\pi(w_{\bullet})\\}}\right)_{i\in
I}=(\phi_{\theta_{0}}\circ\cdots\circ\phi_{\theta_{n}}(p))_{p\in S\wedge
n\in\mathbb{N}\wedge\theta_{0},\ldots,\theta_{n}\in J}$ is
$\mathcal{N}$-summable. We conclude with Proposition 1.26. $\Box$

###  3.5 The Der-Aut correspondence for Noetherian series

Let $\mathbb{V}=k^{[\mathcal{N}]}$ be a space of Noetherian series. By Theorem
3.16, the summability algebra
$k\operatorname{Id}_{\mathbb{V}}+\operatorname{Lin}_{\prec}^{+}(\mathbb{V})$
has evaluations. As a consequence of Corollaries 2.10 and 2.12, we have a
group operation

$\displaystyle\ast:\operatorname{Lin}_{\prec}^{+}(\mathbb{V})\times\operatorname{Lin}_{\prec}^{+}(\mathbb{V})$
$\displaystyle\longrightarrow$
$\displaystyle\operatorname{Lin}_{\prec}^{+}(\mathbb{V})$
$\displaystyle(\phi,\psi)$ $\displaystyle\longmapsto$
$\displaystyle\phi+\psi+\frac{1}{2}(\phi\circ\psi-\psi\circ\phi)+\cdots,$

and a group isomorphism

$\displaystyle\exp:(\operatorname{Lin}_{\prec}^{+}(\mathbb{V}),\ast)$
$\displaystyle\longrightarrow$
$\displaystyle(\operatorname{Id}_{\mathbb{V}}+\operatorname{Lin}_{\prec}^{+}(\mathbb{V}),\circ)$
(12) $\displaystyle\phi$ $\displaystyle\longmapsto$
$\displaystyle\sum_{n\in\mathbb{N}}\frac{1}{n!}\phi^{[n]}.$

Let $\mathbb{A}=k(\\!(M)\\!)$ be an algebra of Noetherian series, where $M$ is
an ordered monoid. Let $1$-$\operatorname{Aut}^{+}_{k}(\mathbb{A})$ denote the
group of bijective strongly linear morphisms of algebra
$\sigma:\mathbb{A}\longrightarrow\mathbb{A}$ which preserve products, and with
$\sigma(a)-a\prec a$ for all $a\in\mathbb{A}\setminus\\{0\\}$. In the case
when $M$ is a linearly ordered group, our notation
$1$-$\operatorname{Aut}^{+}_{k}(\mathbb{A})$ is compatible with that of [19].
Applying Theorem 2.14, we obtain:

######  Theorem 3.17.

The structures $(\operatorname{Der}^{+}_{\prec}(\mathbb{A}),\ast)$ and
$\left(1\mathord{\text{-}}\operatorname{Aut}^{+}_{k}(\mathbb{A}),\circ\right)$
are groups, and we have an isomorphism

$\displaystyle\exp:(\operatorname{Der}^{+}_{\prec}(\mathbb{A}),\ast)$
$\displaystyle\longrightarrow$
$\displaystyle\left(1\mathord{\text{-}}\operatorname{Aut}^{+}_{k}(\mathbb{A}),\circ\right)$
$\displaystyle\partial$ $\displaystyle\longmapsto$
$\displaystyle\sum_{n\in\mathbb{N}}\frac{\partial^{[n]}}{k!}.$

We finish with a formal analog of the Lie homomorphism theorem.

######  Theorem 3.18.

Let $W$ be an ordered set, let $\mathcal{N}_{W}$ be its bornology of
Noetherian subsets and consider the space of Noetherian series
$\mathbb{W}=k^{[\mathcal{N}_{W}]}$. Let
$\Phi:\operatorname{Lin}^{+}_{\prec}(\mathbb{V})\longrightarrow\operatorname{Lin}^{+}_{\prec}(\mathbb{W})$
be a strongly linear morphism of Lie algebras. Then there exists a unique
group morphism
$\Psi:\operatorname{Id}_{\mathbb{V}}+\operatorname{Lin}_{\prec}^{+}(\mathbb{V})\longrightarrow\operatorname{Id}_{\mathbb{W}}+\operatorname{Lin}_{\prec}^{+}(\mathbb{W})$
with

$\Psi(\exp(\phi))=\exp(\Phi(\phi))$

for all $\phi\in\operatorname{Lin}^{+}_{\prec}(\mathbb{V})$.

Proof.  We have
$\operatorname{Id}_{\mathbb{V}}+\operatorname{Lin}_{\prec}^{+}(\mathbb{V})=\exp(\operatorname{Lin}_{\prec}^{+}(\mathbb{V}))$
by (12), so the function $\Psi$ si uniquely determined by

$\Psi(\sigma):=\exp(\Phi(\log(\sigma)))$

for all
$\sigma\in\operatorname{Id}_{\mathbb{V}}+\operatorname{Lin}_{\prec}^{+}(\mathbb{V})$.
Let
$\sigma_{1},\sigma_{2}\in\operatorname{Id}_{\mathbb{V}}+\operatorname{Lin}_{\prec}^{+}(\mathbb{V})$
and write
$(\phi_{1},\phi_{2}):=(\log(\sigma_{1}),\log(\sigma_{2}))\in\operatorname{Der}_{\prec}^{+}(\mathbb{A})$.
Since $\Phi$ is a morphism of Lie algebras, we have
$\Phi(\operatorname{ev}_{(\sigma_{1},\sigma_{2})}(Z_{n}))=\operatorname{ev}_{(\Phi(\sigma_{1}),\Phi(\sigma_{2}))}(K_{n})$
for all $n\in\mathbb{N}$, where $K_{n}\in k\langle\\!\langle
2\rangle\\!\rangle$ is as described in (6). Since $\Phi$ is strongly linear,
we deduce that $\Phi(\operatorname{ev}_{(\phi_{1},\phi_{2})}(X_{0}\ast
X_{1}))=\operatorname{ev}_{(\Phi(\phi_{1}),\Phi(\phi_{2}))}(X_{0}\ast X_{1})$,
i.e.

$\Phi(\phi_{1}\ast\phi_{2})=\Phi(\phi_{1})\ast\Phi(\phi_{2}).$ (13)

It follows that

$\displaystyle\Psi(\sigma_{1}\circ\sigma_{2})$
$\displaystyle=\Psi(\exp(\phi_{1}\ast\phi_{2}))$ (Corollary 2.12)
$\displaystyle=\exp(\Phi(\phi_{1}\ast\phi_{2}))$
$\displaystyle=\exp(\Phi(\phi_{1})\ast\Phi(\phi_{2}))$ (13)
$\displaystyle=\exp(\Phi(\phi_{1}))\circ\exp(\Phi(\phi_{2}))$ (Corollary 2.12)
$\displaystyle=\Psi(\sigma_{1})\circ\Psi(\sigma_{2}).$

Therefore $\Psi$ is a morphism. $\Box$

The same arguments using the identity
$1\text{-$\operatorname{Aut}^{+}_{k}(\mathbb{A})$}=\exp(\operatorname{Der}_{\prec}^{+}(\mathbb{A}))$
give:

######  Theorem 3.19.

Let $\mathbb{B}$ be an angelra of Noetherian series. Let
$\Phi:\operatorname{Der}^{+}_{\prec}(\mathbb{A})\longrightarrow\operatorname{Der}^{+}_{\prec}(\mathbb{B})$
be a strongly linear morphism of Lie algebras. Then there exists a unique
group morphism
$\Psi:1\text{-$\operatorname{Aut}^{+}_{k}(\mathbb{A})$}\longrightarrow
1\text{-$\operatorname{Aut}^{+}_{k}(\mathbb{B})$}$ with

$\Psi(\exp(\partial))=\exp(\Phi(\partial))$

for all $\partial\in\operatorname{Der}^{+}_{\prec}(\mathbb{A})$.

As a consequence of Corollary 2.15, we have:

######  Corollary 3.20.

The group $1\text{-$\operatorname{Aut}^{+}_{k}(\mathbb{A})$}$ is divisible and
torsion-free.

##  4 Toward a full correspondence

###  4.1 Decomposing valuation preserving automorphisms

Let $k$ be an ordered field and let $G$ be a linearly ordered Abelian group.
There is a natural ordering on the field $\mathbb{K}:=k(\\!(G)\\!)$ for which
it is an ordered field [15]. The field $\mathbb{K}$ is a Hahn field as per
[24, 19].

Let $v$-$\operatorname{Aut}^{+}_{k}(\mathbb{K})$ denote the group of strongly
linear automorphisms of the algebra $\mathbb{K}$ with $a\asymp
b\Rightarrow\sigma(a)\asymp\sigma(b)$ for all $a,b\in U(\mathbb{K})$ where
$a\asymp b$ if and only if $a\not\prec b$ and $b\not\prec a$. By [19, Theorem
3.7.1], the group $v$-$\operatorname{Aut}^{+}_{k}(\mathbb{K})$ is the semi-
direct product of the following three subgroups:

  1. a)

The group $1$-$\operatorname{Aut}_{k}^{+}(\mathbb{K})$.

  2. b)

The group $G$-$\operatorname{Exp}(\mathbb{K})$ of functions

$\Psi_{x}\colon a\mapsto\sum_{g\in G}x(g)t^{g}$

where $x\in\operatorname{Hom}((G,+),(k^{\times},\cdot))$.

  3. c)

The group $o$-$\operatorname{Aut}(G)$ of functions

$a\mapsto\sum_{g\in G}a(g)t^{\mu(g)}$

where $\mu\in\operatorname{Aut}(G,+,<)$.

###  4.2 Prelogarithms

Write $\mathbb{K}_{\succ}:=\\{a\in\mathbb{K}\ :\ \operatorname{supp}a<0\\}$.
The ordered group $(\mathbb{K},+,0)$ has [15, Theorems 1.4 and 1.8] an
additive lexicographic decomposition

$\mathbb{K}=\mathbb{K}_{\succ}+k+\mathbb{K}^{\prec},$

whereas $(\mathbb{K}^{\char 62\relax 0},\cdot,1,<)$ has a multipicative
decomposition

$\mathbb{K}^{\char 62\relax 0}=t^{G}\cdot k^{\char 62\relax
0}\cdot(1+\mathbb{K}^{\prec}).$

A prelogarithm is an embedding $\log\colon(\mathbb{K}^{\char 62\relax
0},\cdot,1,<)\longrightarrow(\mathbb{K},+,0,<)$. It is compatible with the
valuation $v$ on $\mathbb{K}$ if for all $a\char 62\relax 0$, we have
$v(\log(a))\geq 0\Longleftrightarrow v(a)\geq 0$ and $v(\log(a))\char 62\relax
0\Longleftrightarrow v(a-1)\char 62\relax 0$. A logarithm is a surjective
prelogarithm. The existence of a compatible logarithm is equivalent [15, Lemma
1.21] to the existence of the three following isomorphisms of ordered groups.

  1. a)

A right logarithm, i.e. an isomorphism
$(1+\mathbb{K}^{\prec},\cdot,1,<)\longrightarrow(\mathbb{K}^{\prec},+,0,<)$.

  2. b)

A middle logarithm, i.e. an isomorphism $(k^{\char 62\relax
0},\cdot,1,<)\longrightarrow(k,+,0,<)$.

  3. c)

A left logarithm, i.e. an isomorphism
$(t^{G},\cdot,1,<)\longrightarrow(\mathbb{K}_{\succ},+,0,<)$.

This is illustrated in the following picture

Although prelogarithms always exist, no right logarithm exsits if $G$ is non-
trivial [18, Theorem 1]. This obstruction can be circumvented by considering
directed unions of fields of Hahn series, such as $\log$-$\exp$ series, or
fields of EL-series [15, Chap 5, Section 2].

One can interpret the formal $\exp$-$\log$ correspondence between each
$\operatorname{Der}_{\prec}^{+}(\mathbb{K})$ and
$1$-$\operatorname{Aut}_{k}^{+}(\mathbb{K})$ as a non-commutative
generalisation of the natural right logarithm on $\mathbb{K}$, which is the
isomorphism

$(1+\mathbb{K}^{\prec},\cdot,<)\longrightarrow(\mathbb{K}^{\prec},+,<)\ ;\
1+\varepsilon\mapsto\sum\limits_{m\in\mathbb{N}}\frac{(-1)^{m}\varepsilon^{m+1}}{m}.$

We want to investigate how this correspondence extends to the group
$v$-$\operatorname{Aut}^{+}_{k}(\mathbb{K})$, mapping it to an appropriate Lie
subalgebra of $\operatorname{Der}^{+}(\mathbb{K})$. Let us first extend the
correspondence to the subgroup
$\operatorname{IntAut}^{+}_{k}(\mathbb{K}):=G$-$\operatorname{Exp}(\mathbb{K})\ltimes
1$-$\operatorname{Aut}^{+}_{k}(\mathbb{K})$ on $\mathbb{K}$.

###  4.3 The middle correspondence

Let $G$-$\operatorname{Der}(\mathbb{K})$ denote the commutative group, under
pointwise sum, of (strongly linear) derivations $\mathrm{d}_{\alpha}$ of the
form

$\forall a\in\mathbb{K},\mathrm{d}_{\alpha}(\sum_{g\in G}a(g)t^{g})=\sum_{g\in
G}\alpha(g)a(g)t^{g}$

where $\alpha\in\operatorname{Hom}((G,+),(k,+))$.

Note that $\operatorname{Hom}((G,+),(k,+))$ and
$G$-$\operatorname{Der}(\mathbb{K})$ are vector spaces and
$\mathrm{d}_{\cdot}$ is a $k$-linear isomorphism. Let
$\mathrm{e}:(k,+,<)\longrightarrow(k^{\char 62\relax 0},\cdot,<)$ be an
isomorphism of ordered groups, i.e. the inverse of a middle logarithm on
$\mathbb{K}$. Writing $k^{\times}$ as the direct product
$k^{\times}\simeq\\{1,-1\\}\times k^{\char 62\relax 0}$, we obtain

$\displaystyle\operatorname{Hom}((G,+),(k^{\times},\cdot))$
$\displaystyle\simeq$
$\displaystyle\operatorname{Hom}((G,+),(\\{1,-1\\},\cdot))\times\operatorname{Hom}((G,+),(k^{\char
62\relax 0},\cdot))$ $\displaystyle\simeq$
$\displaystyle\operatorname{Hom}((G,+),(\mathbb{Z}_{2},+))\times\operatorname{Hom}((G,+),(k,+))$

We have
$\operatorname{Hom}((G,+),(\mathbb{Z}_{2},+))\simeq\operatorname{Hom}((G/2G,+),(\mathbb{Z}_{2},+))$
which is trivial if and only if $G$ is $2$-divisile. In that case, we have an
isomorphism

$\operatorname{Hom}((G,+),(k,+))\longrightarrow\operatorname{Hom}((G,+),(k^{\times},\cdot))\>;\>\alpha\mapsto\mathrm{e}\circ\alpha.$

which yields a Der-Aut correspondence
$G$-$\operatorname{Der}(\mathbb{T})\longrightarrow
G$-$\operatorname{Exp}(\mathbb{T})\ ;\
\mathrm{d}_{\alpha}\mapsto\psi_{\mathrm{e}\circ\alpha}$.

###  4.4 Toward a left correspondence

Consider the Lie algebra $\operatorname{Der}^{+}_{\preceq}(\mathbb{K})$ of
strongly linear derivations
$\partial\colon\mathbb{K}\longrightarrow\mathbb{K}$ such that
$v(\partial(a))\geq v(a)$ for all $a\in\mathbb{K}$. This is the semi-direct
product of the Lie algebra $G$-$\operatorname{Der}(\mathbb{K})$ with the Lie
ideal $\operatorname{Der}^{+}_{\prec}(\mathbb{K})$. Combining the right and
middle $\operatorname{Der}$-$\operatorname{Aut}$ correspondences, we obtain an
isomorphism between $\operatorname{Der}^{+}_{\preceq}(\mathbb{K})$ and the
group $\operatorname{IntAut}_{k}^{+}(\mathbb{K})$ of internal strongly linear
$v$-automorphisms of $\mathbb{K}$. This is summed up in the following picture,
where the upper (resp. lower) decomposition is given by semi-direct internal
products of groups (resp Lie algebras).

What of the left part of the decomposition of
$v$-$\operatorname{Aut}(\mathbb{K})$? Under what conditions on $\mathbb{K}$ is
there a corresponding Lie algebra $\mathbf{D}$ of strongly linear derivations
such that $\operatorname{Der}^{+}_{\preceq}(\mathbb{K})$ is a Lie ideal of
$\mathbf{D}+\operatorname{Der}^{+}_{\preceq}(\mathbb{K})$ and that there
exists a bijective correspondence between $\mathbf{D}$ and
$o$-$\operatorname{Aut}(G)$?

In some cases, automorphisms of $(G,+,<)$ have been shown [16, Proposition
4.9] to induce strongly linear derivations on $\mathbb{K}$. Furthermore, the
derivations can be chosen compatible with a specific prelogarithm on
$\mathbb{K}$ [17, Section 3].

Future work. We plan to investigate this construction method in order to
obtain further derivations coming from automorphisms on $G$.

#### Acknowledgements

This project was initiated at the Fields Institute in June 2022 and resumed at
the Banff International Research Station in February 2024. We wish to thank
both institutions for their hospitality.

Lothar Sebastian Krapp thanks the Vector Stiftung and the Messmer Stiftung,
and Salma Kuhlmann thanks the AFF for travel support.

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V. Bagayoko, Institut de Mathématiques de Jussieu – Paris Rive Gauche, 75013
Paris, France

E-mail address<EMAIL_ADDRESS>
L. S. Krapp, Fachbereich Mathematik und Statistik, Universität Konstanz, 78457
Konstanz, Germany

E-mail address<EMAIL_ADDRESS>
S. Kuhlmann, Fachbereich Mathematik und Statistik, Universität Konstanz, 78457
Konstanz, Germany

E-mail address<EMAIL_ADDRESS>
D. Panazzolo, Université de Haute-Alsace, Laboratoire de Mathématiques,
Informatique et Applications, 68093 Mulhouse, France

E-mail address<EMAIL_ADDRESS>
M. Serra, Fakultät für Mathematik, Technische Universität Dortmund, 44227
Dortmund, Germany

E-mail address<EMAIL_ADDRESS>