# Manin conjecture for statistical pre-Frobenius manifolds, hypercube
relations and motivic Galois group in coding

Noémie C. Combe  Max Planck Institute for Mathematics in the Sciences,
Inselstr. 22,04103, Leipzig  To Yuri Ivanovitch Manin on the occasion of his
birthday with admiration and gratefulness. Thank you so much for teaching me
so many incredible things and for opening my mind to a new world.

(Paganini, Capriccio 3 for violin solo, Presto<EMAIL_ADDRESS>
###### Abstract.

This article develops, via the perspective of (arithmetic) algebraic geometry
and category theory, different aspects of geometry of information. First, we
describe in the terms of Eilenberg–Moore algebras over a Giry monad, the
collection $Cap_{n}$ of all probability distributions on the measurable space
$(\Omega_{n},\mathcal{A})$ (where $\Omega$ is discrete with $n$ issues) and it
turns out that there exists an embedding relation of Segre type among the
product of $Cap_{n}$’s. We unravel hidden symmetries of these type of
embeddings and show that there exists a hypercubic relation. Secondly, we show
that the Manin conjecture—initially defined concerning the diophantine
geometry of Fano varieties—is true in the case of exponential statistical
manifolds, defined over a discrete sample space. Thirdly, we introduce a
modified version of the parenthesised braids (${\bf mPaB}$), which forms a key
tool in code-correction. This modified version ${\bf mPaB}$ presents all types
of mistakes that could occur during a transmission process. We show that the
standard parenthesised braids ${\bf PaB}$ form a full subcategory of ${\bf
mPaB}$. We discuss the role of the Grothendieck–Teichmüller group in relation
to the modified parenthesised braids. Finally, we prove that the motivic
Galois group is contained in the automorphism $Aut(\widehat{{\bf mPaB}}).$ We
conclude by presenting an open question concerning rational points,
Commutative Moufang Loops and information geometry.

###### Key words and phrases:

Exponential varieties, Moufang loops, toric varieties, Manin conjecture,
Grothendieck–Teichmüller group, Segre embedding

I acknowledge the Minerva Grant from the Max Planck Society for supporting my
work.

###### Contents

  1. 0 Introduction
  2. 1 Statistical pre-Frobenius manifolds in relation to algebraic geometry and Manin’s conjecture
  3. 2 Words, codes and algebraic structures in information transmission
    1. 3 Modified parenthesised Braids as a key to code-correction

##  0\. Introduction

The terminology geometry of information refers to models of databases subject
to noise. This connects to quantification, storage, and communication of
digital information. Applications of fundamental topics of information theory
include source coding, data compression, and channel coding or error detection
and correction. In this paper, we consider algebraic structures occurring in
geometry of information and we prove surprising connections between the theory
of geometry of information and diophantine geometry (see for instance [CMS20]
for an introduction).

We unravel tight bridges between objects of information geometry (such as
manifolds of probability distributions and codes) and diophantine geometry and
algebraic geometry. In the first part of this paper, we work with the space of
probability distributions on finite sets (see [Ama10] for an introduction and
[CPN21, CCN22, CM20] for new developments). Probability distributions are used
in many problems such as machine learning, vision, statistical inference,
neural networks and others, this development provides a strong tool for many
areas. We consider the class of statistical manifolds of exponential type. The
aim of the first part of the paper is to consider the structure of the space
of probability distributions and the category of these spaces.

Our work is subdivided into three main parts. The first part regards the
collection of all probability distributions on the measurable (discrete with
$n$ issues) space $Cap_{n}$ and their embeddings into $Cap_{m}$ (where $m>n$).
It turns out that a hidden hypercubic symmetries appear. The second part,
proves that the Manin conjecture concerning rational points on a Fano variety
can be extended to the case of information geometry. The second part, regards
a codes/ error-correcting codes aspect of information geometry and we show a
tight relation to the motivic Galois group and a modified version of
parenthesised braids, serving as a way of encoding all possible errors
occurring a given word.

In the first part, we give a proof of the following statement:

######  Theorem.

(Thm. 1.1.1) Let $Cap_{n}=Cap(\Omega_{n},\mathcal{A}_{n})$ be the collection
of all probability distributions on the measurable (discrete) space
$(\Omega_{n},\mathcal{A})$ where $\Omega$ is formed from $n$ outcomes. Then,
the diagram of all embeddings of multi-product of
$\underbrace{Cap_{2}\times\cdots\times Cap_{2}}_{n+1\,times}$ in
$Cap(\Omega_{2^{n+1}},\mathcal{A}_{2^{n+1}})$ has the structure of an
$n$-cube.

In the second part of this paper, we show an extension of the Manin conjecture
to a wider family of objects, showing thus deep connections between
information geometry and arithmetics/ algebraic geometry. In particular, we
show that Manin’s conjecture concerning the diophantine geometry of Fano
varieties [FMT89] holds in the case of exponential statistical manifolds,
defined over a discrete sample space. Initially, the Manin conjecture states
the following.

“Let $V$ be a Fano variety (defined over a number field $K$); let $H$ be a
height function, relative to the anticanonical divisor, and assume that $V(K)$
is Zariski dense in $V$. Then there exists a non-empty Zariski open subset
$U\subset V$ such that the counting function of $K$-rational points of bounded
height, defined by the set $N_{{U,H}}(B)=\\#\\{x\in U(K):H(x)\leq B\\}$ for
$B\geq 1$, satisfies the relation $N_{{U,H}}(B)\sim cB(\log B)^{{\rho-1}}$, as
$B\to\infty$, where $c$ is a constant.” A reformulation of this statement in
terms of the pre-Frobenius statistical manifolds of exponential type (defined
on a discrete sample space) is given. We show that this conjecture is true for
those manifolds of information geometry.

######  Theorem.

(Thm. 1.2.1) Consider $S=\\{p(q,\theta)\\}$ an exponential statistical
manifold (over a discrete sample space $(\Omega,\mathcal{A})$ of finite
dimension.

  * •

Let $T=(\mathbb{Q}^{*})^{m}$ be the $\mathbb{Q}$-torus of the exponential
statistical manifold given by the probability coordinates.

  * •

Consider $\mathbb{P}_{\Sigma}$ the smooth $\mathbb{Q}$-compactification of the
torus $T$ i.e. a smooth, projective $\mathbb{Q}$-variety in which $T$ lies as
a dense open set and $\Sigma$ is a Galois invariant regular complete fan.

  * •

Let $k$ be the rank of the Picard group $Pic(\mathbb{P}_{\Sigma})$.

Then, there is only a finite number $N(T,\mathcal{K}^{-1},B)$ of
$\mathbb{Q}$-rational points $x\in T(\mathbb{Q})$ having the anticanonical
height $H_{\mathcal{K}^{-1}}(x)\leq B$. Moreover, as $B\to\infty$:

$N(T,\mathcal{K}^{-1},B)=\frac{\Theta(\Sigma)}{(k-1)!}\cdot
B(logB)^{k-1}(1+o(1)),$

where $\Theta(\Sigma)$ is a constant.

In former works of Manin and collaborators [CMM21a, CMM21b], was shown the
existence of Moufang patterns encoding various symmetries appearing naturally
in models, related to storing and transmitting information such as information
spaces. By Moufang patterns we have in mind in particular loops (such as
Moufang loops). The latter form non-associative analogs of groups. For the
case of spaces of probability distributions on finite sets the symmetries of
these spaces have the structure of Commutative Moufang Loops.

Loops and quasigroups turn out to play a central role when it comes to
considering geometry of information. The aspect relating geometry of
information and (virtually) non-commutative Moufang Loops appears in the
context of error-correcting codes and algebraic-geometry codes [CMM21a].

In the second part of this work, we consider an aspect of geometry of
information directly related to semantics and to the theory of error-
correcting codes and to errors [MM11, MM16]. Any natural language can be
considered as a tool for producing large databases. Communication (or
transmission of information) refers to the process by which a sender
communicates a message (i.e. a union of sequences of letters forming words
defined in a given alphabet and associated to a given meaning) to a receiver.
During a given communication the message can arrive distorted. We investigate
the cases where the message is subject to distortion (or is coded) and arrives
finally modified.

A modification can take different aspects such as: permutations of letters,
replacement of a letter by another one, removal of a letter or on the contrary
extension of words by adding letters and sequences of words. In particular,
when it comes to coding, latin squares can be used to decode the message.

We consider the space of all possible modifications of a words (indexed by
their length) and suppose that our words are parenthesised. Suppose that we
have a pair of parenthesised words $w$ and $w^{\prime}$ of the same length. It
turns out that an object which is perfect for the description of this
situation and which also offers a geometric vision of paths of errors made
during a transmission is tightly related to the groupoid of parenthesised
braids, introduced to study the Grothendieck–Teichmüller group and which was
first considered by Drinfeld.

In order to cover all sorts of error transmissions we introduce an enriched
version of the groupoid of parenthesised braids, denoted ${\bf mPaB}$ for
modified parenthesised braids. This modification of the classical
parenthesised braid is necessary due to the fact that the words are allowed to
have repeating letters and this is strongly related to loops and quasigroups.
The braids are consequently impacted. Thus we equip the standard braids with
two supplementary operations called pinching and attaching operations.

This groupoid of modified parenthesised braids inherits naturally the
operations of cabling $d_{i}$, strand removal $s_{i}$, extension $d_{0}$ as
well as $\square$, the coproduct functor defined by setting each individual
parenthesised braid $B$ to be group-like, i.e. $\square(B)=B\otimes B$ and
$\sigma$ the elementary braid on two strands.

It turns out that the there is a strong connection between the space of all
possible transmission errors and arithmetics. Indeed the pro-unipotent
Grothendieck–Teichmüller group (and thus the motivic Galois group) are
included in the automorphism of the pro-unipotent completion of $\widehat{{\bf
mPaB}}$.

######  Theorem.

(Thm. 3.2.1 and Cor. 3.2.1) The motivic Galois group is contained in the
automorphism of the pro-unipotent completion of the modified parenthesised
braids $Aut(\widehat{{\bf mPaB}})$.

To conclude, we discuss an open question relating the Commutative Moufang
Loops (CML) structure arising in the symmetries of the spaces of probability
distributions on a discrete set. The appearance of the simplest CML’s in the
algebraic-geometric setup is motivated by smooth cubic curves in a projective
plane $\mathbb{P}_{K}^{2}$ over a field $K$. It is in particular shown that
the set $E$ of $K$-points of such a curve $X$ forms a CML. Regarding the
result of the first part of the paper, we conjecture that the set $E$ of
$\mathbb{Q}$-points of a pre-Frobenius statistical manifold forms also a CML.

### Plan of the paper

-Sec.1. of the paper is devoted to considering $Cap_{n}$ which is the collection of all probability distributions on the measurable space $(\Omega_{n},\mathcal{A})$. The sample space is discrete and $\Omega_{n}$ has $n$ outputs. There exists an associated monad (called the Giry monad) and an algebra over it (Eilenberg–Moore algebra). We discuss in particular the relation among the product of $Cap_{n}$, which turns out to be hypercubic. Secondly, we prove that the the Manin conjecture holds for (pre-Frobenius) the statistical manifolds related to exponential families and defined over a discrete (finite) set (Sec.1.2).

-Sec. 2 we discuss another aspect that geometry of information can take, via codes and error codes. It serves as an intermezzo between Sec. 1 and the Sec. 3. and prepares the ground for what follows. After recalling definitions on Loops and quasigroups we study in particular the algebraic properties of the space of modified words and show that quasigroups and loops offer a perfect set up for this.

-Sec. 3 we introduce our modified parenthesised braids, which forms a key tool in code-correction. We show that the standard parenthesised braids ${\bf PaB}$ are a full subcategory of ${\bf mPaB}$. We discuss the role of the Grothendieck–Teichmüller group in relation to the modified parenthesised braids (Sec.3.2). Finally, we end the section by showing that the motivic Galois group is contained in the automorphism $Aut(\widehat{{\bf mPaB}}).$ We conclude finally by presenting an open question concerning rational points, Commutative Moufang Loops and information geometry 3.3.

##  1\. Statistical pre-Frobenius manifolds in relation to algebraic geometry
and Manin’s conjecture

###  1.1. Categorical introduction of considered objects

Dealing with classical information theory leads to working in the following
framework. Let $(\Omega,\mathcal{A},P_{0})$ a probability space where $\Omega$
is the space of elementary outputs, $\mathcal{A}$ is the $\sigma$-algebra of
events and $P_{0}$ is a probability measure, (usually $P_{0}\ll\mu$,
$P_{0}[A]=\int_{A}f_{0}d\mu$ for some ($\sigma$-finite) measure $\mu$).

– The algebra $\mathfrak{B}_{c}$ is the (commutative) algebra of all bounded
measurable functions $f$ on the space of elementary outcomes $\Omega$.

$\mathfrak{B}_{c}$ algebra with respect of addition and multiplication by a
scalar of function $f:\omega\to\mathbb{R},\quad f^{-1}(x)\in\mathcal{A},\quad
x\in\mathbb{R}$

– The probability state of an object is determined by a nonnegative,
normalized, normal (i.e., ultra- weakly continuous, or what is the same,
monotone continuous) linear functional $\Phi_{c}$ one, $\mathfrak{B}_{c}$,
$\Phi_{c}:\mathfrak{B}_{c}\to\mathbb{R}$, which is the expectation with
respect to some probability $P_{0}$:

$\Phi_{c}(f)=\mathbb{E}_{P_{0}}[f].$

– The set $\mathfrak{I}(\mathfrak{B}_{c})$ of all states $\Phi_{c}$ of an
object is a convex closed set in the pre-dual space $\mathfrak{B}_{c}$,
$(\mathfrak{B}_{c})^{\star}=\mathfrak{B}_{c}$.

– The idempotents of $\mathfrak{B}_{c}$ are just the indicators
(characteristic functions) of the measurable sets (elements of $\mathcal{A}$),
these subspaces are called events (or “yes-no” experiments).

Before we enter a categorical definition, let us mention that the collection
of all probability measures on a measurable space $(\Omega,\mathcal{A})$ of
elementary outcomes is a convex subset of the semi-ordered linear space of
measures of bounded variations on $(\Omega,\mathcal{A})$. In some cases, it
useful to remark that the collection of all probability measures on
$(\Omega,\mathcal{A})$ is equipped with a norm, giving rise to a metric space.
However, this aspect will not be important to us here.

These measures are endowed with the supplementary property that they are
invariant under maps of the collection of probability measures induced by
invertible measurable maps of the sample space $(\Omega,\mathcal{A})$. This
means that given a pair of sample spaces $(\Omega_{1},\mathcal{A}_{1})$ and
$(\Omega_{2},\mathcal{A}_{2})$ equipped with their corresponding collection of
probability measures, say $\\{P_{\theta}^{i},\,\theta\in\Theta\\}$ where
$i\in{1,2}$ and with (same) parameter set $\Theta\in\mathbb{R}^{n}$ one can
develop a notion of equivalence: $\\{P_{\theta}^{1},\,\theta\in\Theta\\}$ and
$\\{P_{\theta}^{2},\,\theta\in\Theta\\}$ are said to be equivalent whenever
there exist Markov maps $\Pi^{12}$ and $\Pi^{21}$ such that
$P^{1}_{\theta}\Pi^{12}=P^{2}_{\theta}$ and
$P^{2}_{\theta}\Pi^{21}=P^{1}_{\theta}$, for any $\theta\in\Theta$.

We call $Cap$ the collection of all probability distributions on the
measurable space $(\Omega,\mathcal{A}).$ The discussion above leads to
defining a category denoted $CAP$, where objects are isomorphic classes of
collections of all probability distributions on the measurable spaces
$(\Omega,\mathcal{A})$; morphisms are given by the Markov maps. These Markov
maps correspond to statistical decision rules in the sense of Wald.

Further algebraic operations on the objects of the category are allowed and
are defined as follows. One can define a direct product of measurable spaces.
This construction implies the existence of a tensor product on the collection
of all probability measures on those measurable spaces. This multiplication is
functorial with respect to the Markov category.

To give an example, let us take $Cap_{2}$, where
$\Omega=\\{\omega_{1},\omega_{2}\\}$ and the probability distributions
$p=\langle p_{1},p_{2}\rangle.$ Defining $Cap_{2}\times Cap_{2}$ is given by
$\\{\omega_{1},\omega_{2}\\}\times\\{\omega^{\prime}_{1},\omega^{\prime}_{2}\\}$
and this corresponds to
$\\{\omega_{1}\omega^{\prime}_{1},\omega_{1}\omega^{\prime}_{2},\omega^{\prime}_{1}\omega_{1},\omega^{\prime}_{2}\omega_{1}\\}$,
which can be rewritten as:
$\\{\omega^{{}^{\prime\prime}}_{1},\omega^{{}^{\prime\prime}}_{2},\omega^{{}^{\prime\prime}}_{3},\omega^{{}^{\prime\prime}}_{4}\\}$.
If $p=\langle p_{1},p_{2}\rangle$ and $q=\langle q_{1},q_{2}\rangle$ are the
corresponding probability distributions then the tensor product on the
probability distributions is such that:

$p\otimes q=\langle p_{1}q_{1},p_{1}q_{2},p_{2}q_{1},p_{2}q_{2}\rangle.$

We shall investigate more precisely what happens during this multiplication
process in $Cap_{n}$. In particular we show, using Segre type of embeddings
that we have a hypercube type of relation within these operations.

However, note that in this paper, we are not interested in the quantum aspect
of information theory and we limit ourselves to the case where the algebra is
commutative. Hence, we do not consider the quantum information geometry aspect
which requires a von Neumann algebra $\mathfrak{b}$ of bounded linear
operators acting on Hilbert space.

This algebra $\mathfrak{b}$ corresponds to a (generally non-commutative)
analogue of the classical commutative algebra of all bounded measurable
functions on the space of elementary outcomes. The Hermitian elements of the
algebra $\mathfrak{b}$ are called bounded observables. The probability state
of an object is determined by a (nonnegative, normalised, normal, monotone
continuous) linear functional $\phi$ on the algebra $\mathfrak{b}$. The set of
all states $S(\mathfrak{b})$ of a given object forms a convex closed set in
the pre-dual space $\mathfrak{b}_{*}$, where
$(\mathfrak{b}_{*})^{*}=\mathfrak{b}$.

Moreover, analogous constructions concerning the Markov maps can be defined
and the system of all Markov maps of all collections $S(\mathfrak{b})$ forms
an algebraic category.

To summarise, there exist tight relations between convex sets and the sets of
states in the probabilistic computation (discrete or continuous) and in
quantum computation. We will explore this from a more categorically aspect. In
particular, we invoke the Giry monad structure and define an Eilenberg–Moore
algebra over this monad to consider our convex sets for information geometry.

Consider the category ${\rm Set}$ of finite sets. Given an object $X$ of ${\rm
Set}$, we define

$\Delta_{X}=\big{\\{}\tilde{f}:X\to[0,1]\,|\,\tilde{f}\,\text{has finite
support and}\,\sum_{x\in X}\tilde{f}(x)=1$

the simplex over $X$. It is the set of formal finite convex combinations of
elements from $X$. Elements of $\Delta_{X}$ are the discrete probability
distributions over $X$. The mapping from the set $X$ to the set $\Delta_{X}$
can be made functorial (known as the simplex functor) and defined such that
given a morphism of sets $f:X\to Y$ one defines
$\Delta(f):\Delta_{X}\to\Delta_{Y}$. One may define, for these convex sets,
the algebraic structure of a monad $(\Delta,\mu,\eta)$, where the unit is
given by $\eta:X\to\Delta_{X}$ and the multiplication is defined by
$\mu:\Delta_{X}^{2}\to\Delta_{X}$. This monad is commutative. Moreover, given
an object $X\in Ob({\rm Set})$ and the structure map $\gamma:\Delta_{X}\to X$
commutativity for the following diagrams is satisfied:

${X}$${X}$${\Delta_{X}}$$\scriptstyle{\eta_{X}}$$\scriptstyle{Id_{X}}$$\scriptstyle{\gamma}$${\Delta_{\Delta_{X}}}$${\Delta_{X}}$${\Delta_{X}}$${X.}$$\scriptstyle{\mu_{X}}$$\scriptstyle{\Delta_{\gamma}}$$\scriptstyle{\gamma}$$\scriptstyle{\gamma}$

Taking a pair $(X,\gamma)$ allows to work in an Eilenberg–Moore algebra for
the distribution monad $(\Delta,\mu,\eta)$ over the symmetric monoidal
category of sets. An algebra morphism
$f:(X,\gamma)\to(X^{\prime},\gamma^{\prime})$ is a continuous map such that
the following diagram commutes:

${\Delta_{X}}$${X}$${\Delta_{X^{\prime}}}$${X^{\prime}.}$$\scriptstyle{\Delta(f)}$$\scriptstyle{\gamma}$$\scriptstyle{f}$$\scriptstyle{\gamma^{\prime}}$

An Eilenberg–Moore algebra of the monad $(\Delta,\mu,\eta)$ is a map of the
form $\gamma:\Delta_{X}\to X$ s.t. $\gamma\circ\eta=id$ and
$\gamma\circ\mu=\gamma\circ\Delta_{\gamma}.$ Note that each category of
algebras for a monad on sets is cocomplete. Regarding the category of
Eilenberg–Moore algebras it is both complete and cocomplete.

A more global idea hides behind this, in the context of probability
distributions and concerning the relation between algebras for the Giry monad
[Gir82] and the convex spaces formed by the collection of probability
distributions. This comes from the following statement:

The category of algebras for the Giry monad is isomorphic to the category of
$G$-partitions. Here by $G$-partition we mean that for $X$ (in a fully general
setting $X$ is a polish space, i.e. a separable metric space for which a
complete metric exists) corresponds to a collection $\\{G(x)\,|\,x\in X\\}$,
which forms a positive convex partition for $\Delta$ into closed sets indexed
by $X$. Moreover, $\delta_{x}\in G(x)$ (where $\delta_{x}$ is the Dirac
measure on $x$) holds for all $x\in X$, and the set valued map $x\mapsto G(x)$
is $k$-upper-semicontinuous.

Now, algebras over a commutative monad admit a tensor product. By $\star$ we
denote the monoidal multiplication of $\Delta$. So, given $\Delta$-algebras
$(A,a)$ and $(B,b)$ their tensor product is the object $A\otimes_{\Delta}B$
given by:

$\Delta(\Delta A\otimes\Delta B)\xrightarrow{\star}\Delta\Delta(A\otimes
B)\xrightarrow{\mu}\Delta(A\otimes B)\to A\otimes_{\Delta}B.$

Now that we have explicitly shown the algebraic structure of the convex spaces
of probability distributions, and discussed the tensor product operation for
algebras over the Giry monad, we are interested in proving the existence of
hidden symmetries appearing within the multiplication relations between
$Cap_{n}$’s. In particular, this leads to proving the existence of a hypercube
relation.

Let us go back to the previous example $Cap_{2}\times Cap_{2}\hookrightarrow
Cap_{4}$. Recall the tensor product relation on the probability distributions
“à la Morozova–Chentsov” [MC91], where one considers the probability
distributions under the shape of a vector in an affine space. Regarding our
example, this gives us:

$p\otimes q=\langle p_{0}q_{0},p_{0}q_{1},p_{1}q_{0},p_{1}q_{1}\rangle,$

so that we can consider $p\otimes q$ as the 4-tuple: $\langle
p_{0}q_{0},p_{0}q_{1},p_{1}q_{0},p_{1}q_{1}\rangle=\langle
p^{\prime}_{0},p^{\prime}_{1},p^{\prime}_{2},p^{\prime}_{3}\rangle$ defined in
the affine space. This relation can obviously be generalised for any $n$ i.e
for $Cap_{n}$.

Now, from an affine $n$-tuple $p=\langle p_{1},\cdots,p_{n}\rangle$ one can
take easily the homogeneous coordinates: $P=[p_{1}:p_{2}:\cdots:p_{n}]$. So,
now whenever we consider the product $P\otimes Q$, where we consider it from
the projective perspective, one has $P=[p_{0}:p_{1}:\cdots:p_{n}]$ and
$Q=[q_{0}:q_{1}:\cdots:q_{m}]$. This leads to defining a (real) Segre
embedding, which looks as follows:

$h:\mathbb{P}^{n}\times\mathbb{P}^{m}\hookrightarrow\mathbb{P}^{(n+1)(m+1)-1},$

where $\mathbb{P}^{n}$ is the real projective space.

Taking a pair of points $(P,Q)$ in the projective space
$\mathbb{P}^{n}\times\mathbb{P}^{m}$ (corresponding to elements in $Cap$) one
can define the product:

$([p_{0}:p_{1}:\cdots p_{n}],[q_{0}:q_{1}:\cdots
q_{m}])\mapsto[p_{0}q_{0}:p_{0}q_{1}:\cdots:p_{i}q_{j}:\cdots p_{n}q_{m}].$

In particular, going back to the $Cap_{2}$ case, with $P=[p_{0}:p_{1}]$ (resp.
$Q=[q_{0}:q_{1}]$), where $0\leq p_{i}\leq 1$ and $p_{0}+p_{1}=1$ (resp.
$0\leq q_{i}\leq 1$ and $q_{0}+q_{1}=1$) we have the following commutative
diagram:

${\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}}$${\mathbb{P}^{3}\times\mathbb{P}^{1}}$${\mathbb{P}^{1}\times\mathbb{P}^{3}}$${\mathbb{P}^{7}}$$\scriptstyle{Id\times
h_{(23)}}$$\scriptstyle{h_{(12)}\times
Id}$$\scriptstyle{h_{((12)3)}}$$\scriptstyle{h_{1(23)}}$

The following embedding

$\underbrace{\mathbb{P}^{1}\times\cdots\times\mathbb{P}^{1}}_{n\
times}\to\mathbb{P}^{2^{n}-1}$

corresponds to a generalised Segre embedding. This allows to consider the
relation between $Cap_{2}$ and $Cap_{2^{n}}$ and leads to the following remark
on the geometry of $Cap_{2^{n}}$.

######  Remark 1.1.1.

Note that this highlights a new way to show the existence of a paracomplex
structure on objects in $Cap$. Note that a paracomplex projective space of
dimension $n$ is identified to a product of projective spaces of the same
dimension $n$ (and defined over the real numbers) i.e.
$\mathbb{P}^{n}\times\mathbb{P}^{n}$. Since, we can use the Segre embedding
and the construction above we can see that for instance in $Cap_{4}$ we have
an embedded paracomplex projective space. Using the generalised Segre
embedding, the statement generalises for $Cap_{2^{n}}$. The paracomplex
structure has been mentioned in the work [CM20]. However, this gives another
approach to that result.

The generalised Segre embedding implies the existence of $(n-1)$-hypercube
relations when considering $\underbrace{Cap_{2}\times\cdots\times
Cap_{2}}_{n\,times}.$ Let us start with the following relation.

$\underbrace{\mathbb{P}^{1}\times\cdots\times\mathbb{P}^{1}}_{n\
times}\to\mathbb{P}^{2^{n}-1}.$

We proceed by analogy on $Cap_{n}$ so that we can in fact obtain a similar
relation as in the Segre emebedding:

$\underbrace{Cap_{2}\times\cdots\times Cap_{2}}_{n\,times}\to Cap_{2^{n}}$

Take $n=3$. For the product $Cap_{2}\times Cap_{2}\times Cap_{2}$ we have that
the following square commutative diagram:

${\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}}$${\mathbb{P}^{3}\times\mathbb{P}^{1}}$${\mathbb{P}^{1}\times\mathbb{P}^{3}}$${\mathbb{P}^{7}}$$\scriptstyle{h_{(12)}\times
Id_{3}}$$\scriptstyle{Id_{1}\times
h_{23}}$$\scriptstyle{h_{(12)3}}$$\scriptstyle{h_{1(23)}}$

######  Theorem 1.1.1.

The diagram of embeddings of $\underbrace{Cap_{2}\times\cdots\times
Cap_{2}}_{n+1}$ in $Cap_{2^{n+1}}$ has the structure of an $n$-cube.

###### Proof.

The proof can be done by using a bijection between the set of vertices and
edges of the generalised Segre embedding diagram and the set of vertices/
edges constructing the $n$-cube.

Take a product
$\underbrace{\mathbb{P}^{1}\times\cdots\times\mathbb{P}^{1}}_{n+1}$. For any
pair of (adjacent) projective spaces in this cartesian product, to which the
Segre embedding is applied, add a pair of parenthesis (one open and one
closed) such that

$(\mathbb{P}^{1}\times\mathbb{P}^{1}\times\cdots\underbrace{(\mathbb{P}^{1}\times\mathbb{P}^{1})}_{i,i+1}\cdots\times\mathbb{P}^{1})\hookrightarrow(\mathbb{P}^{1}\times\mathbb{P}^{1}\times\cdots\underbrace{\mathbb{P}^{3}}_{i}\cdots\times\mathbb{P}^{1}).$

The construction goes as follows and it goes by induction on $n\geq 1$.

  * •

To any parenthesised product
$\underbrace{(\mathbb{P}^{1}\times\mathbb{P}^{1})\times\mathbb{P}^{1}\times\cdots\times(\mathbb{P}^{1}\times\mathbb{P}^{1})}_{n+1}$
corresponds a binary word $x=(x_{1},\cdots,x_{n})$ with $n$ letters such that
$x_{i}\in\\{0,1\\}$, such that
$\underbrace{\mathbb{P}^{1}\overbrace{\times}^{(x_{1},}\mathbb{P}^{1}\overbrace{\times}^{x_{2},}\mathbb{P}^{1}\cdots\overbrace{\times}^{\cdots,x_{n})}\mathbb{P}^{1}}_{n+1}$
where:

$\begin{cases}x_{i}=0&\text{if there exists {\bf no} parenthesis for the
pair}(i,i+1)\text{of projective spaces:}\\\
&\underbrace{\mathbb{P}^{1}\times\cdots\overbrace{\mathbb{P}^{1}\times\mathbb{P}^{1}}^{i,i+1}\times\mathbb{P}^{1}}_{n+1}.\\\
x_{j}=1&\text{if there exists a parenthesis for the
pair}\underbrace{\mathbb{P}^{1}\times\cdots\overbrace{(\mathbb{P}^{1}\times\mathbb{P}^{1})}^{j,j+1}\times\mathbb{P}^{1}}_{n+1}\\\
\end{cases}$

So the number of parenthesis is given by the number of units in the word. For
each unit added, the remaining zeros of the word correspond to the remaining
projective spaces which have not yet been paired.

  * •

for any $n\geq 1$, the product
$\underbrace{\mathbb{P}^{1}\times\cdots\times\mathbb{P}^{1}}_{n+1}$
corresponds to $\underbrace{(0,\cdots,0)}_{n}$.

  * •

$(1,1,\cdots,1)$ corresponds to the projective space $\mathbb{P}^{2^{n+1}-1}$.

  * •

Each combination of parenthesis being encoded by a binary word
$(x_{1},\cdots,x_{n})$ corresponds to the vertex of the Segre embedding
diagram.

  * •

Add one parenthesis to a given combination. This corresponds to adding a unit
to the word i.e. $x_{j}=1$ if we have
$\mathbb{P}^{1}\times\mathbb{P}^{1}\times\cdots\underbrace{(\mathbb{P}^{1}\times\mathbb{P}^{1})}_{j,j+1}\cdots\times\mathbb{P}^{1}$
at the $j$-th and $(j+1)$-th position.

  * •

An edge of the diagram is drawn whenever to words $x$ and $x^{\prime}$ differ
only by one letter i.e. there exists one unique $j$ such that $x_{j}\neq
x^{\prime}_{j}$. For $i\neq j$ we have $x_{i}=x^{\prime}_{j}$.

$\star$ Let us discuss the low dimensional case. Take
$\mathbb{P}^{1}\times\mathbb{P}^{1}$. The corresponding word has one letter
$(x_{1})$. This corresponds to $Cap_{2}\times Cap_{2}$. Then, by the Segre
embedding $(\mathbb{P}^{1}\times\mathbb{P}^{1})\hookrightarrow\mathbb{P}^{3}$.
The new vertex obtained by the Segre embedding modifies the word 0 into the
word 1. The diagram is just a segment (so a cube of diemsion 1).

$\star$ We have $\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$. Let
us use our construction, where vertices of the embedding diagram are encoded
by the binary words of length 2, i.e. $(x_{1},x_{2})$ where
$x_{i}\in\\{0,1\\}.$ The initial vertex is encoded by
$\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$ which corresponds to
the word $(0,0)$. Two vertices are connected by an edge whenever the pair of
corresponding words differ by only one character. So, for instance, the vertex
(0,0) is directly connected by an edge to the vertices $(0,1)$ and $(1,0)$ but
not connected to the vertex word (1,1). The diagram is a square.

$\star$ For the case $Cap_{2}\times Cap_{2}\times Cap_{2}\times Cap_{2}$, it
is easy to check that one has a cube diagram relation.

$\star$ In full generality, the relations of embeddings of
$\underbrace{\mathbb{P}^{1}\times\cdots\times\mathbb{P}^{1}}_{n+1\ times}$ in
the generalised Segre embedding have the structure of a $n$-hypercube graph.

In fact, this statement follows from the definition of a hypercube (or
$n$-cube) which is a graph of order $2^{n}$, whose vertices are represented by
$n$-tuples $(x_{1},\cdots x_{n})$ where $x_{i}\in\\{0,1\\}$ and whose edges
connect vertices which differ in exactly one term. We use the construction
above, where we have established a bijection between the set of vertices
indexed by binary words and the parenthesised product of projective spaces;
edges of the $n$-cube correspond to applying one Segre embedding to a pair of
parenthesised projective spaces.

So, to conclude we have a hypercube graph relation illustrating the diagram of
relations in $Cap_{n}$. ∎

We illustrate a four dimensional cube (a tesseract) in the figure below (Fig.
1), where vertices are indexed by words of length 4 and letters are in
$\\{0,1\\}$. Using the above construction, we can exactly illustrate the Segre
embedding relations for :

$\underbrace{Cap_{2}\times\cdots\times Cap_{2}}_{5\,times}\to Cap_{32},$

which in a projective version corresponds to illustrating
$\underbrace{\mathbb{P}^{1}\times\cdots\times\mathbb{P}^{1}}_{5\,times}\to\mathbb{P}^{31}.$

$\scriptstyle(1111)$$\scriptstyle\mathbb{P}^{31}$$\scriptstyle(0110)$$\scriptstyle(1001)$$\scriptstyle(0000)$$\scriptstyle\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$$\scriptstyle(0111)$$\scriptstyle(1110)$$\scriptstyle(0001)$$\scriptstyle(1000)$$\scriptstyle(0011)$$\scriptstyle(1100)$$\scriptstyle(0101)$$\scriptstyle(1010)$$\scriptstyle(1011)$$\scriptstyle(0100)$$\scriptstyle(0010)$$\scriptstyle(1101)$

Figure 1: Segre diagram for
$\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$
with vertices labeled by binary words (commas have been omitted for
simplicity).

###  1.2. Manin’s conjecture theorem for discrete exponential families

The theory of exponential varieties reveals the existence of a surprisingly
strong connection to diophantine geometry. We show that the asymptotic formula
conjectured by Manin, for the case of Fano varieties, concerning the number of
$K$-rational points of bounded height with respect to the anticanonical line
bundle holds in the case of a smooth projectivisation of an exponential
variety (defined for a discrete finite sample space). This statement extends
to the framework of information geometry the conjectured by Manin, which
initially was stated in the context of algebraic geometry.

Regarding the previous subsection, we are now working on an object of the
category $Cap$. Our statement goes as follows:

######  Theorem 1.2.1.

Consider $S=\\{p(q,\theta)\\}$ an exponential statistical manifold (over a
discrete sample space $(\Omega,\mathcal{A})$) of finite dimension.

  * •

Let $T=(\mathbb{Q}^{*})^{m}$ be the $\mathbb{Q}$-torus of the exponential
statistical manifold given by the probability coordinates.

  * •

Consider $\mathbb{P}_{\Sigma}$ the smooth $\mathbb{Q}$-compactification of the
torus $T$ i.e. a smooth, projective $\mathbb{Q}$-variety in which $T$ lies as
a dense open set and $\Sigma$ is a Galois invariant regular complete fan.

  * •

Let $k$ be the rank of the Picard group $Pic(\mathbb{P}_{\Sigma})$.

Then, there is only a finite number $N(T,\mathcal{K}^{-1},B)$ of
$\mathbb{Q}$-rational points $x\in T(\mathbb{Q})$ having the anticanonical
height $H_{\mathcal{K}^{-1}}(x)\leq B$. Moreover, as $B\to\infty$:

$N(T,\mathcal{K}^{-1},B)=\frac{\Theta(\Sigma)}{(k-1)!}\cdot
B(logB)^{k-1}(1+o(1)),$

where $\Theta(\Sigma)$ is a constant.

######  Remark 1.2.1.

The exponential statistical manifold is a pre-Frobenius manifold and we will
can refer to it as the pre-Frobenius statistical manifold (for a definition of
pre-Frobenius manifold we refer from instance to [Man99]).

The proof of this statement is done in two parts. The first is to state
explicitly the relation from exponential varieties (defined as above for
finite, discrete sample space) to toric varieties. The second part is to apply
the theorem of Batyrev–Tschinkel in this context.

###  1.3. Exponential statistical manifolds for discrete sample space

A statistical variety (or manifold) can be considered as the parametrized
family of probability distributions $S=\\{p(x;\theta)\\}$, where
$p(x;\theta)=\frac{dP_{\theta}}{d\mu}$ is the Radon–Nikodym derivative of
$P_{\theta}$ w.r.t. the $\sigma$-finite measure $\mu$ (and it is positive
$\mu$-almost everywhere). It comes equipped with the following ingredients:

  * •

the canonical parameters:
$\theta=(\theta^{1},\dots,\theta^{n})\in\mathbb{R}^{n}$;

  * •

the symbol $x$ referring to a family of random variables $\\{x_{i}\\}$ on a
sample space $\Omega$;

  * •

$p(x;\theta)$ is the probability distribution parametrized by $\theta$.

A family $S=\\{p(x;\theta)\\}$ of distributions is an exponential family if
the density functions can be written in the following way:

$p(x;\theta)=\exp(\theta^{i}x_{i}-\Psi(\theta)),$

where

  * •

$\Psi(\theta)$ is a potential function, which is given by
$\Psi(\theta)=\log\int_{\Omega}\exp\\{\theta^{i}x_{i}\\}d\mu$.

  * •

the parameter $\theta$ and $x=(x_{i})_{i\in I}$ (where $I$ is a finite set)
have been chosen adequately;

  * •

the canonical parameter satisfies
$\Theta:=\\{\theta\in\mathbb{R}^{d}:\Psi(\theta)<\infty\\}$.

Whenever $p(x;\theta)$ is smooth enough in $\theta$ one can include in the
statistical model a structure of an $n$-dimensional manifold. We use the
construction of the family $A$ as a manifold, using the atlas
$\\{U_{i},\phi_{i}\\}_{i\in I}$.

From now on suppose that the sample space is finite and discrete, i.e.
$\Omega=\\{\omega_{1},\cdots,\omega_{m}\\}$. A small change of notation is
required for practical reasons. This leads us to consider the exponential
family of probability distribution defined by:

(1)
$p(q;\theta)=p_{0}(\omega)\exp\\{\theta^{i}q_{i}(\omega)-\Psi(\theta)\\},\,\text{where}\quad
p_{0}(\omega)>0$

and with canonical distribution parameter
$\theta=(\theta^{1},\cdots,\theta^{n})\in\mathbb{R}^{n}$. Furthermore, we have
$\omega\in\Omega$ elements of the sample space and $q_{i}:\Omega\to\mathbb{R}$
are a family $q=\\{q_{i}\\}$ of random variables; $\Psi(\theta)$ is a cumulant
generating function. The $q_{i}(\omega),i\in I$ ($I$ is some list of indices)
is a function defining directions of the coordinate axes, called statistics
(or directional sufficient statistics).

######  Proposition 1.3.1.

The exponential statistical manifolds (defined as above and over a discrete
and finite sample space) have the structure of a real toric variety.

We present the construction below.

###### Proof.

$\bullet$ Let us define $\mathcal{Q}=\\{\bf{q}_{1},\cdots,\bf{q}_{n}\\}$,
where ${\bf q_{i}}=(q_{i1},\cdots,q_{im})^{T}\subset\mathbb{Z}^{m}$ and the
components satisfy $q_{ij}:=q_{i}(\omega_{j})$, with $\omega_{j}\in\Omega$.
The matrix $\mathcal{Q}$ has size $m\times n$ and its components are integers.
Columns are given by the set $\\{{\bf q_{1}},\cdots,{\bf q_{n}}\\}$. The
matrix $\mathcal{Q}=[q_{ij}]_{i=1,\cdots,n;\,j=1,\cdots,m}$, where
$q_{ij}=q_{i}(\omega_{j})$ and $q_{0}(\omega_{j})=1$ give the directional
statistics.

$\bullet$ Put $t_{i}=e^{\theta^{i}}\in\mathbb{R}_{>}$. The monomial is then
$t_{i}^{q_{i}(\omega_{j})}=\exp{\\{\theta_{i}q_{i}(\omega_{j})\\}}$ and one
can write the following equation:

$\exp\left\\{\sum_{i=1}^{n}\theta^{i}q_{ij}\right\\}=\prod_{i=1}^{n}t_{i}^{q_{ij}}=\boldsymbol{\tau_{j}}$

So, to conclude, we have that $p(q;t)$ can be rewritten as the product
$t_{1}^{{\bf q}_{1}}\cdots t_{n}^{{\bf q}_{n}}$.

Moreover, since we assumed that $q_{ij}$ are integers, $\boldsymbol{\tau_{j}}$
(for $j=1,\cdots,m$) form Laurent polynomials in $t_{i}$. Therefore, each
vector ${\bf q_{i}}$ is identified to a monomial $t^{\bf q_{i}}$ in the
Laurent polynomial ring $\mathbb{Q}[t^{\pm}]$, where
$\mathbb{Q}[t^{\pm}]:=\mathbb{Q}[t_{1},\cdots,t_{m},t_{1}^{-1},\cdots,t_{m}^{-1}]$.

Statistically speaking, the monomial $t_{i}^{q_{i}(\omega_{j})}$ can be
interpreted as the probability of having the canonical parameter $\theta_{i}$
in the direction of $q_{i}(\omega_{j})$ for the event $\omega_{j}.$ Whereas,
the $m$-tuple $(t_{1},\cdots,t_{m})\in(\mathbb{Q}^{*})^{m}$, where
$t_{i}=\exp{\theta^{i}}$ form the probability coordinates.

Going back to the classical construction of the toric ideal, we apply the
following. Take the (semigroup) homomorphism:

$\pi:\mathbb{N}^{n}\to\mathbb{Z}^{m},\quad\mathbf{u}=(u_{1},\cdots,u_{n})\mapsto\sum_{i=1}^{n}u_{i}{\bf
q_{i}}.$

The image of $\pi$ is the semigroup:

$\mathbb{N}\mathcal{Q}=\\{\lambda_{1}{\bf q_{1}}+\cdots\lambda_{n}{\bf
q_{n}}\,:\,\lambda_{1},\cdots,\lambda_{n}\in\mathbb{N}\\}.$

This map $\pi$ lifts to a homomorphism of semigroup algebras:

$\hat{\pi}:\mathbb{Q}[{\bf y}]\to\mathbb{Q}[t^{\pm 1}],\quad y_{i}\mapsto
t^{\bf q_{i}},$

where $\mathbb{Q}[{\bf y}]$ is a polynomial ring in the variables ${\bf
y}:=(y_{1},\cdots,y_{n})$.

It is the kernel of the homomorphism $\hat{\pi}$ that generates the toric
ideal $\mathcal{I}_{T}$ of $\mathcal{Q}$. The multiplicative group
$(\mathbb{Q}^{*})^{m}$ is known as the $m$-dimensional algebraic torus. The
variety of the form $V(\mathcal{I}_{T})$ is the affine toric variety. So, we
have shown the existence of an $m$-dimensional algebraic torus for the
exponential statistical manifolds. This algebraic torus is given by the
probability coordinates $(t_{1},\cdots,t_{m})\in(\mathbb{Q}^{*})^{m}$, where
$t_{i}=\exp{\theta^{i}}$.

Note that for $dim(\mathcal{Q})=m$, one can visualise the dense torus using
the fact that the set $V(\mathcal{I}_{T})\cap(\overline{\mathbb{Q}}^{*})^{m}$
is an algebraic group under coordinate-wise multiplication which is isomorphic
to the $m$-dimensional torus $T=(\overline{\mathbb{Q}}^{*})^{m}$.

To each point $P\in U_{i}$, where $(U_{i},\phi_{i})$ is a chart, we apply the
homomorphism construction above. The coordinate functions $y_{i}$ on the chart
$U_{i}$ can be expressed as Laurent monomials in the adequate coordinates. In
changing from one chart to another the coordinate transformation remains
monomial. So, this forms a smooth toric variety, where we have a collection of
charts $y_{i}:U_{i}\to\mathbb{Q}^{n}$, such that on the intersections of
$U_{i}$ with $U_{j}$ the coordinates $y_{i}$ must be Laurent monomials in
$y_{j}$.

A toric variety with a collection of charts determines a system of cones
$\\{\sigma_{a}\\}$ in $\mathbb{R}^{n}$. Putting coordinates $x_{1},\cdots
x_{n}$ on a given fixed chart $U_{0}$ the coordinate functions $x^{(a)}$ on
the remaining charts $U_{a}$ can be represented as Laurent monomials in
$x_{1},\cdots x_{n}$.

Furthermore, if we have a regular function $f$ on $U_{a}$, then it can be
represented as a Laurent polynomial in $x_{1},\cdots,x_{n}$. The regularity
condition for a function $f$ on the chart $U_{a}$ can be expressed in terms of
the support of the corresponding Laurent polynomial $\tilde{f}.$ For
$\tilde{f}=\sum_{m\in\mathbb{Z}}c_{m}x^{m}$ the support of $\tilde{f}$ is the
set $\\{m\in\mathbb{Z}^{n}\,|\,c_{m}\neq 0\\}$ and with each chart $U_{a}$, we
associate a cone $\sigma_{a}$ generated by the exponents of
$x_{1}^{(a)},\cdots,x_{n}^{(a)}$ as Laurent polynomials in $x_{1},\cdots
x_{n}$.

An arbitrary Laurent polynomial $\tilde{f}$ is regarded as a rational function
on $X$. Regularity of this function on the chart $U_{a}$ is equivalent to
$supp(\tilde{f})\subset\sigma_{a}$. Thus, various questions on the rational
function $\tilde{f}$ on the toric variety $X$ reduces to the combinatorics of
the positioning of $supp(\tilde{f})$ with respect to the system of cones
$\\{\sigma_{a}\\}$.

Reciprocally, one can construct a toric variety by specifying a system of
cones $\\{\sigma_{a}\\}$ satisfying certain properties. These requirements can
be most conveniently stated in terms of the system of dual cones and leads to
the notion of fan. ∎

######  Corollary 1.3.1.

Consider the exponential statistical variety defined for a discrete finite
sample space. If we have a regular function $f$ on $U_{a}$, then it can be
represented as a Laurent polynomial in $x_{1},\cdots,x_{n}$. The regularity
condition for a function $f$ on the chart $U_{a}$ can be expressed in terms of
the support of the corresponding Laurent polynomial $\tilde{f}$ and in the
exponential variety it is given by the directional statistics. In particular,
the cone is generated by the directional statistics.

###### Proof.

This follows from the discussion above. ∎

Now, we argue that the Manin conjecture holds for exponential statistical
manifolds. Indeed, following the construction of Batyrev–Tschinkel [BT98], the
Manin conjecture is true for toric varieties. From the above statement (Prop.
1.3.1) it follows that the exponential statistical manifolds (defined over
finite sample space) have the structure of a (real) toric variety. Therefore,
the conclusion follows easily that a smooth projectivised version of the
exponential statistical manifolds defined over finite and discrete sample
space satisfies the Manin conjecture.

##  2\. Words, codes and algebraic structures in information transmission

###  2.1. Motivation

As was shown in previous works, Moufang loops and quasigroups are central in
information geometry. We focus on the situation of coding or of error making
during a transmission of a given information. It turns out that the algebraic
structures of loops and quasigroups offers the right language and formalism to
deal with this type of problems. This is starting to be developed in Sec. 2.3
and the following sections. We recall below some results relating structures
codes and non-necessarily commutative Moufang loops and quaisgroups.

Commutative Moufang Loops appear in the symmetries of the space of
probabilities: automorphisms of order two that are boundary limits of the
reflections of geodesics about the center, come equipped with a structure of a
quasigroup. These automorphisms define a composition law on the set of points
that forms an abelian quasigroup.

Similarly, non-necessarily commutative Moufang loops and quasigroups appear
among the other aspect of information geometry, regrouping around codes/
structure codes (see [CMM21a, CMM21b]). We will mention a few results in
relation to this in what follows.

Based on the works in [MM11], family of codes are defined as follows. We
choose and fix an integer $q\geq 2$ and a finite set: the alphabet $A$ of
cardinality $q$. An (unstructured) code $C$ is defined as a nonempty subset
$C\subset S$ of words of length $n\geq 1$. The sequence $w=(\alpha_{i})$ of
elements of $A$, where $i=1,2,...,n$ is called a word $w$ of length $n$. We
denote by $n(C)$ the common length of all words in $C$. Such a subset $C$
comes equipped with its code point datum. This is given by a pair $P_{C}=({\rm
R}(C),\delta(C))$, where ${\rm R}(C)$ is called the transmission rate and
$\delta(C)$ is the relative minimal distance of the code.

  * •

The relative minimal distance of the code $\delta(C)$ is given by the quotient
$\delta(C):=\frac{d(C)}{n(C)},$ where $d(C)=\min\\{d(a,b)\,|\,a,b\in C,a\neq
b\\}$ is the minimal distance between two different words in $C$; $n(C):=n$
and $d(a,b)$ is the Hamming distance between two words:
$d((\alpha_{i}),(\alpha^{\prime}_{i})):=card\\{i\in(1,\cdots,n)\,|\,\alpha_{i}\neq\alpha^{\prime}_{i}\\};$

  * •

The transmission rate ${\rm R}(C)$ depends on the $log_{q}(Card(C))$ i.e. we
have: ${\rm R}(C)=\frac{[log_{q}(Card(C))]}{n(C)}$.

Note that for our investigations the code point $P_{C}=({\rm R}(C),\delta(C))$
will not be directly considered, although it is implicitly present.

As mentioned in earlier works of [CMM21a] (section 5.2), Moufang symmetries
generally become visible in the so-called structured codes. The most studied
structure codes appear in linear codes and algebraic-geometric codes.
Concerning the former (linear codes) one considers the alphabet
$S:=\mathbb{F}_{q},$ corresponding to generators of a finite field of
cardinality $q$, and $C\subset\mathbb{F}_{nq}$ form $\mathbb{F}_{q}$-linear
subspaces. Concerning the latter (algebraic-geometric codes) one has the same
class of alphabets, but the difference is that one considers
$\mathbb{F}_{q}$-points in an affine (or projective) $\mathbb{F}_{q}$-scheme
with a chosen coordinate system.

Moufang symmetries appear indirectly in this geometric setting. Their
existence can be seen using various and different formalisms, motivated for
instance by theoretical physics. Let us recall some definitions on loops and
quasigroups.

###  2.2. Quasigroups and Moufang symmetries

For the convenience of the reader, we recall below the algebraic structures of
quasigroups, loops, Moufang loops.

  1. (1)

Let $A$ be a finite set of cardinality $q$. A binary operation on a set $A$ is
a mapping $\diamond:A\times A\to A$ which associates to every ordered pair
$(a,b)$ of elements in $A$ a unique element $a\diamond b$. A set with a binary
operation is called a magma.

  2. (2)

A quasigroup is a magma (i.e. a set $A$ with a binary multiplication denoted
by $\diamond$) such that in the equation $x\diamond y=z$ the knowledge of any
two of $x,y,z$ specifies uniquely the third. Latin squares form the
multiplication tables of quasigroups.

  3. (3)

Based upon the set $A$ a Latin square is a $|A|\times|A|$ array in which each
element of $A$ occurs exactly once in each row and exactly once in each
column. In particular, for all ordered pairs $(a,b)\in A^{2}$ there exist
unique solutions $x,y\in A$ to the equations: $x\diamond a=b,\quad a\diamond
y=b$, and those solutions are precisely given by the Latin squares.
Differently speaking, for each element $r$ of a magma $(A,\diamond)$ one can
define the left multiplication:

$L(r)=L_{x}(r):A\to A,x\mapsto r\diamond x$

and the right multiplication:

$R(r)=R_{x}(r):A\to A,x\mapsto x\diamond r.$

The operators $L$ and $R$ form bijections of the underlying set $A$. We call
them left (resp. right) translation maps. In particular, this allows to
reformulate the definition of the quasigroup, using the translation maps so
that a magma is a combinatorial quasigroup iff the left multiplication $L(r)$
and the right multiplication $R(r)$ are bijective for each element $r$ of $A$.

We can add to this structure the possibility of having a unit denoted $e$ i.e.
such that $e\diamond a=a\diamond e=a$ holds for any element $a\in S$.

  4. (4)

A quasigroup $(A,\diamond)$ is a nonempty set $A$ equipped with a binary
multiplication $\diamond:A\times A\to A$ and such that, for each $a\in S$, the
right and left translation maps $R(a):A\to A$ and $L(a):A\to A$ given by
$R(a)=r\diamond a$ and $L(a)=a\diamond r$, are permutations of $A$. If there
is a two-sided identity element $1_{A}=1_{(A,\diamond)}$ then $A$ is a loop.

  5. (5)

A loop is a called Moufang if it is a unital quasigroup (it has a unit and
every element is invertible) with a near associativity relation:

$(a\diamond b)\diamond(c\diamond d)=a\diamond((b\diamond c)d),$
$a\diamond(a\diamond b)=(a\diamond a)\diamond b,\quad(a\diamond
b)\circ(a\diamond c)=(a\diamond a)\diamond(b\diamond c).$

where $(a,b,c,d)\in S^{4}$.

Going back to our previous discussion on code loops, the loop $\mathcal{L}$
is, roughly speaking, given by the sequence:

$0\to R\to\mathcal{L}\to C\to 0,$

where $R$ is a ring (which will be more precisely defined below);
$C\subset\mathbb{F}^{n}_{2^{r}}$ is a linear code, equipped with an additional
structure which is introduced in the next paragraph: the “almost-symplectic
structure”. In order to give a flavour to the reader we recall this notion and
expose how the loops appear in more details. For further information we refer
to [MM16].

An almost symplectic structure on a finite dimensional vector space $V$ over
$\mathbb{F}_{q}$ ($q$ odd) is a non-degenerate skew-symmetric form
$\omega:V\times V\to\mathbb{F}_{q}$ where $\omega$ satisfies the anti-symmetry
$\omega(u,v)=-\omega(v,u)$, with $\omega(u,0)=\omega(0,u)=0$, and for any non-
null element $u$ in $V$ there exists some $v\in V$ satisfying $\omega(u,v)\neq
0$. A polarisation of the almost-symplectic form is a function $\beta:V\times
V\to R$ satisfying the relation

$\beta(u,v)-\beta(v,u)=\omega(u,v).$

Consider the finite field $\mathbb{F}_{2^{r}}$ and identify it to the residue
field: $\mathcal{O}_{K}/{\bf m}_{K}$, where $K$ is an unramified extension of
degree $r$ of ${\bf Q}_{2}$; $\mathcal{O}_{K}$ is the ring of integers and
${\bf m}_{K}$ the maximal ideal. The ring $R$ in the above short exact
sequence is given by $R=\mathcal{O}_{K}/{\bf m}^{2}_{K}$.

The construction of the almost-symplectic code loop $\mathcal{L}(V,\beta)$
over $\mathbb{F}_{q}$ where $q=2^{r}$ is an extension given by the short exact
sequence:

$0\to R\to\mathcal{L}(V,\beta)\to V\to 0,$

where $(V,\beta)$ is an almost-symplectic vector space $(V,\omega)$ with
polarization $\beta$ over $\mathbb{F}^{n}_{2^{r}}$.

This setup motivates our investigations concerning codes and error-codes. In
particular, we give a construction allowing to take into account all possible
errors (or error corrections) occurring during the transmission of some
information. The framework of quasigroups and loops fits adequately this type
of problem.

###  2.3. Words, codes and algebraic structures

The algebraic structure of spaces of words and codes are interesting to study.
As soon as one associates to words of the code $C$ some given meaning, a code
$C$ forms a type of dictionary for a given language. A finite combination of
code words form sentences in this language. However, it can happen that given
an information encoded by such a sentence it might be distorted during the
transmission and so mistakes may appear in the receivers message, changing
thus its meaning.

The types of mistakes that can possibly occur are listed below:

  1. (1)

letters in a word can be permuted,

  2. (2)

one letter can be replaced by another letter (we say that this letter has been
translated or shifted to another one),

  3. (3)

new letters can be added to the word,

  4. (4)

letters can be lost in the word,

  5. (5)

new words can be added to the preexisting word.

In the following part of this section we consider the first two types of
mistakes. We argue that quasigroups and loops offer the perfect setting to
define these types of operations. Mistakes of type (3), (4), (5) are
considered in the next section where the notion of modified parenthesised
braids is introduced.

######  Example 1.

Consider the alphabet $A_{4}=\\{a,b,c,d\\}$ and suppose the Latin square
associated is as follows.

$\cdot$ | $a$ | $b$ | $c$ | $d$
---|---|---|---|---
$a$ | $b$ | $c$ | $d$ | $a$
$b$ | $c$ | $d$ | $a$ | $b$
$c$ | $d$ | $a$ | $b$ | $c$
$d$ | $a$ | $b$ | $c$ | $d$

Then the word $w=(c(ab)d)$ can be distorted using the translation maps as
$(L_{a}(c)(aR_{b}(b))d)$ and the receiver reads $(d(ad)d)$. We do not assume
commutativity (unless it is clearly stated) i.e. the word bac is not
equivalent to cab. When it comes to parenthesised words, associativity is not
allowed either, so that $b(ac)$ is not equivalent to $(ba)c$.

We now introduce the following notations and explicit the corresponding
notions.

  * •

Consider an alphabet $A$ (finite set of cardinality $n$).

  * •

By $M_{p}(A)$ we denote the parenthesised $p$-words formed from the alphabet
$A$. Repetitions of letters are allowed.

  * •

$M_{n}(A)$: sum of $M_{p}(A)\times M_{n-p}(A)$, where $1\leq p\leq n-1$. A
word $w\in M_{n}(A)$ can be written as the concatenation of two smaller words,
strictly contained in between an open and a closed parenthesis i.e. we have
$w=(w^{\prime})\circ(w^{\prime\prime})$ where $w^{\prime}$ is of length $p$
and $w^{\prime\prime}$ of length $n-p$. We call those subwords the blocks of
$w$.

  * •

The sum of the family $(M_{n}(A))_{n\geq 1}$ is denoted $M(A)$.

  * •

$M(A)$: the free magma, with composition law $w,w^{\prime}\mapsto w\circ
w^{\prime}$.

  * •

$\mathbb{M}_{k}(A)$ is used only for parenthesised $k$-words with distinct
letters. Note that for this notation to be consistent it is necessary that
$k\leq n$. In particular, $\mathbb{M}_{n}(A)$ are the parenthesised $n$-words
with $n$ distinct letters.

There is a clear separation of $M_{n}(A)$ into two subclasses made of those
words with distinct letters $\mathbb{M}_{n}(A)$ and those words with repeating
letters.

######  Remark 2.3.1.

Concerning the last class of parenthesised words, if we take for example
$A=\\{a,b,c,d\\}$, then an element of $\mathbb{M}_{5}A$ forms a word where
letters repeat: the expression $(a(bd))(ba)$ represents an element of
$\mathbb{M}_{5}A$.

We now consider the connection between the free magma structure $M(A)$ and the
magma $(A,\circ)$ on which the quasigroup acts $GQ=(A,\diamond,L,R)$. Suppose
for simplicity that $A$ has cardinality $n$. Then we have an action on the
sequence of letters forming a word $w=(x_{1}\cdots x_{n})$ that we can write
as an $n$-tuple i.e. $(x_{1},\cdots,x_{n})$ such that each entry (letter) of
the n-tuple is translated by a left $L$ or right map $R$. To avoid any source
of confusion we denote the translation of all letters of the word as $T$. So,
we have the following:

$QG\times M_{n}(A)\to M_{n}(A)$
$(T,(x_{1},\cdots,x_{n}))\to(T(x_{1}),\cdots,T(x_{n}))$

Recall the construction of $M(A)$. For any $w\in M(A)$, there exists a unique
$n\geq 1$ such that $w\in M_{n}(A).$ For a given pair of words
$(w,w^{\prime})\in M_{p}(A)\times M_{q}(A)$ of length $p$ and $q$ respectively
we can define a product $w\circ w^{\prime}$ forming an element of
$M_{p+q}(A)$. The set $M(A)$ with the law composition $w,w^{\prime}\mapsto
w\circ w^{\prime}$ is the free magma. So, we can state the following lemma.

######  Lemma 2.3.1.

Let $A_{n}$ be a finite set of cardinality $n$. Consider the magma
$\mathfrak{N}:=(A_{n},\circ)$ and suppose that there exists a quasigroup
$QG_{n}=(A_{n},\diamond,R,L)$ acting on the letters of the words of length
$n$. Then, there exists a unique morphism $g:M(A)\to\mathfrak{N}$ from the
free magma on $A_{n}$ to $\mathfrak{N}$.

###### Proof.

A quasigroup is a magma $\mathfrak{N}$ where every element is invertible. Let
us define the following bijection $f:A\to\mathfrak{N}$. By induction we can
construct the morphism $g$ as follows.

  * •

Let $f_{1}=f:M_{1}(A)\to\mathfrak{N}$, where $M_{1}(A)=A$.

  * •

For $n\geq 2$, we have $f_{n}:M_{n}(A)\to\mathfrak{N}$ and given $w\circ
w^{\prime}\in M_{p}(A)\times M_{n-p}(A)$, $f_{n}(w\circ
w^{\prime})=f_{p}(w)\circ f_{n-p}(w^{\prime})$.

There exists only one unique morphism $g$ inducing $f_{n}$ on $M_{n}(A)$ for
all $n\geq 1$. So, $g$ is the unique morphism of $M(A)$ into $\mathfrak{N}$
which extends $f$. ∎

######  Lemma 2.3.2.

Consider the quasigroup $(A,\diamond,L,R)$ acting on $M_{n}(A)$, $n\geq 2$.
Then, any permutation of the letters of a word $w\in M_{n}(A)$ can be
recovered by an adequate combination and choice of translation maps $L$ and
$R$.

###### Proof.

Any permutation can be obtained by a product of transpositions. Now, the
operators $L_{x}$ and $R_{y}$ define a transposition iff

(2) $x\diamond a=b\quad\text{and}\quad b\diamond y=a,$

where $x,y\in A$. So, given a quasigroup $(A,\diamond,L,R)$ any permutation of
the letters of a word $w\in M_{n}(A)$ can be obtained for every $n\geq 2$ from
$L$ and $R$. ∎

Restricting our attention to $\mathbb{M}_{n}A$, we have the following free
magma structure, defined as follows:

  * •

$\mathbb{M}_{0}A=\emptyset$

  * •

$\mathbb{M}_{1}A=A$

  * •

$\mathbb{M}_{n}A=\sqcup_{p+q=n}\mathbb{M}_{p}A\times\mathbb{M}_{q}A.$

######  Remark 2.3.2.

We can interpret $\mathbb{M}_{n}A$ differently as a set of rooted binary
planar trees (each vertex has exactly two incoming edges) with $n$ leaves
labelled by elements of $A$.

Let $M=\\{M(n)\\}$ be the symmetric sequence where $M(n)$ is the subset of
$\mathbb{M}_{n}\\{1,\cdots,n\\}$ consisting of the monomials in
$\\{1,...,n\\}$ where each element of the set occurs exactly once. The
symmetric group $\mathbb{S}_{n}$ acts from the right on $M(n)$ by permuting
the elements of the set $\\{1,...,n\\}$. The symmetric sequence $M$ becomes an
operad with operadic composition given by replacing letters by monomials (or
grafting binary trees). The operad $M$ is called the magma operad.

######  Corollary 2.3.1.

Let $A_{n}$ be a set of cardinality $n$. Let $M=\\{M(n)\\}$ be the symmetric
sequence, where $M(n)$ is the subset of $\mathbb{M}_{n}\\{A_{n}\\}$. Then, any
permuted $n$-sequence of $M(n)$ in $(A_{n})^{n}$ can be recovered from the
action of the quasigroup $(A_{n},\diamond,L,R)$ on the elements of
$\mathbb{M}_{n}\\{A_{n}\\}$ where for any transposition $(x_{i}x_{j})$ we put
the condition that $L(x_{i})=x^{\prime}_{i}$ and $R(x_{j})=x^{\prime}_{j}$ for
$x^{\prime}_{i}=x_{j}$ (resp. $x^{\prime}_{j}=x_{i}$) and the rest of the
letters remain unchanged.

###### Proof.

Every $M(n)$ is formed by all words of length $n$, where letters are all
distinct. As was previously shown any permutation of a pair of letters in a
word $w\in M(n)$ is obtained from $L$ and $R$ by applying condition 2. So, any
symmetric sequence in $M(n)$ is obtained by taking a word with distinct
letters and one can apply the operators $L$ and $R$ (and condition (2)) to any
pairs of letters so as this defines a transposition. So, the action of the
quasigroup $(A_{n},\diamond,L,R)$ on $n$-sized words with distinct letters
where $n\geq 1$ allows the construction of any element in $M(n)$. ∎

######  Definition 1.

The elements in $M(n)=(\mathbb{M}_{n}(A_{n}),\mathbb{S}_{n})$ are called
symmetric sequences of length $n$; whereas sequences of length $n$ defined
from the alphabet $A_{n}$ and carrying an action of a quasigroup on $A_{n}$
are translated sequences of length $n$ and denoted $M_{QG}(n)$

Their relation can be described in the next diagram:

${M(n)}$${M_{QG}(n)}$${M(A_{n})}$${(A_{n},\circ)}$$\scriptstyle{i}$$\scriptstyle{j}$$\scriptstyle{f_{n}}$$\scriptstyle{g}$

In short, we can consider two different objects, one being contained in the
other one. The first one is given by the collection of symmetric sequences
$\\{M(n)\\}_{n\geq 1}$. Morphisms between elements of $M(n)$ are given by the
right action of the symmetric group $\mathbb{S}_{n}$. The second object is
given by the collection $\\{M(A_{n})\\}_{n\geq 1}$ where morphisms are
translation maps, generated by $L$ and $R$ acting componentwise on the
$n$-tuples forming $n$-sized words of $M(A_{n})$.

can be obtained from translated sequences. All these structures can be
obtained set theoretically from the free magma. The free magma allows a
decomposition of the magma $(A_{n},\circ)$ by word length.

So, if we restrict our considerations to the case, where translation maps form
permutations, the following diagram appears:

${\mathbb{M}_{n}(A_{n})}$${M_{n}(A_{n})}$${M(A_{n})}$${(A_{n},\circ)}$${(A_{n}^{\sigma},\circ)}$${\mathbb{M}_{n}(A_{n}^{\sigma})}$${M_{n}(A_{n}^{\sigma})}$${M(A_{n}^{\sigma})}$$\scriptstyle{\cong}$$\scriptstyle{i}$$\scriptstyle{f_{n}}$$\scriptstyle{j}$$\scriptstyle{g}$$\scriptstyle{\cong}$$\scriptstyle{\sigma\in\mathbb{S}_{n}}$$\scriptstyle{i}$$\scriptstyle{f_{n}^{\sigma}}$$\scriptstyle{j}$

where we have the inclusion morphisms $i,j$. The inclusion $j$ goes from the
degree $n$ component generated by all words of length $n$ to the free magma;
$A_{n}^{\sigma}=\sigma(A_{n})$, and $\sigma\in\mathbb{S}_{n}$ is a permutation
obtained from and right combination of translation maps $L$ and $R$ such that
they satisfy condition (2) for any pair of transpositions.

######  Lemma 2.3.3.

The composition of translation maps acting on a set $A_{n}$ is associative.

For a word of length 3 one can for instance take: Given the following data:

  * •

Word: $(x_{1},x_{2},x_{3})$

  * •

$f=((L_{a},R_{b},L_{c}),(x_{1},x_{2},x_{3}))$

  * •

$g=((R_{a},R_{c},L_{b}),(y_{1},y_{2},y_{3}))$

  * •

$h=((L_{b},L_{c},R_{b}),(z_{1},z_{2},z_{3}))$

$LHS=((h\circ g)\circ
f)(x)=(L_{b}(R_{a}(y_{1})),L_{c}(R_{c}(y_{2})),R_{b}(L_{b}(y_{3})))\circ
f(x)=$
$(L_{b}(R_{a}(L_{a}(x_{1}))),L_{c}(R_{c}(R_{b}(x_{2}))),R_{b}(L_{b}(L_{c}(x_{3}))))$
$RHS=(h\circ(g\circ
f)(x)=h\circ(R_{a}(L_{a}(x_{1})),R_{c}(R_{b}(x_{2})),L_{b}(L_{c}(x_{3})))=$
$L_{b}(R_{a}(L_{a}(x_{1})),L_{c}(R_{c}(R_{b}(x_{2}))),R_{b}(L_{b}(L_{c}(x_{3})))$

So, RHS=LHS.

###### Proof.

One can easily check that for a word of length $n$ the statement is true using
induction. ∎

######  Lemma 2.3.4.

Consider a (possibly non reduced) Latin square associated to a quasigroup
$(A_{n},\diamond,L,R)$, such that the first row (resp. column) corresponds to
the sequence of the letters of a word $w\in\mathbb{M}_{n}(A_{n})$. Then the
$n-1$ other rows (resp. columns) of the Latin square form sequences of
$n$-words being a permutation of $w$.

###### Proof.

Take an $n$-word with $n$ distinct letters, $w\in\mathbb{M}_{n}(A_{n})$. We
use (a possibly non reduced version of) the Latin square, such that the
sequence of letters in the word $w$ forms the first row or first column. The
multiplication table, forming the latin square, gives permutations of the word
$w$. Multiplying each letter of the word $w$ by an element $a\in A_{n}$ gives
a new row or column indexed by the element $a$. ∎

######  Example 2.

Using the previously discussed example, it is easy to check the above lemma by
taking on the first row the monomial $(badc)$. It gives a non reduced Latin
square, described below.

$\cdot$ | $b$ | $a$ | $d$ | $c$
---|---|---|---|---
$b$ | $d$ | $c$ | $b$ | $a$
$a$ | $c$ | $b$ | $a$ | $d$
$d$ | $b$ | $a$ | $d$ | $c$
$c$ | $a$ | $d$ | $c$ | $b$

It is easy to see that applying the translation maps to the entire word $w$
gives three other permutations: $cbad$, for $L_{a}$; $dcba$ for $L_{b}$ and
$adcb$ for $L_{c}$.

##  3\. Modified parenthesised Braids as a key to code-correction

In this section, we introduced an object that we call the modified
parenthesised Braids. This object serves as a model to consider the space of
all paths of errors that may occur during the transmission of a given
information. A particular advantage of this object is that it helps visualise
the distorsion process easily (via modified braids) and thus leads to more
facility for the correction process.

Previously, we have shown that for all $n\geq 1$, each $M_{n}(A)$ comes
equipped with a decomposition:

$M_{n}(A)=\sqcup_{1\leq p\leq n-1}M_{p}(A)\times M_{n-p}(A),$

indexed by the partitions of the integer $n$.

This precise procedure allows to define the parenthesis in the case of
parenthesised words. The number of ways to insert $n$ pairs of parentheses in
a word of $n+1$ letters is the celebrated Catalan number $Cat(n)$. For $n=2$
there are 2 ways: $((ab)c)$ or $(a(bc))$, as for $n=3$ there are 5 ways:
$((ab)(cd)),(((ab)c)d),((a(bc))d),$ $(a((bc)d)),(a(b(cd)))$. This is in
bijection with all the binomial$(2n,n)$ paths on $\mathbb{Z}$ lattice that
start at $(0,0)$ end at $(2n,0)$, where each step corresponds either to making
a $(+1,+1)$ step or a $(+1,-1)$ step. The number of such paths that never go
below the $x$-axis (also known as the Dyck paths) is $C(n)$.

######  Proposition 3.0.1.

Let $n\geq 1$ be an integer. To every parenthesised word in $w\in
M_{n}(A_{n})$ one can construct a corresponding Dyck path of size $n$ in the
real plane starting at $(0,0)$ ending at $(2n,0)$.

###### Proof.

Using the inductive construction on $M_{n}(A_{n})$ mentioned above, it is easy
to obtain a word with parenthesis. Now, concerning the Dyck paths, a step up
i.e. with coordinates $(+1,+1)$ correspond to an opening of parenthesis and a
step down (i.e. step with $(+1,-1)$ coordinates) corresponds to a closing
parenthesis. Some vertices of the Dyck path may carry a label which
corresponds to the letter(s) of the corresponding block in the word. ∎

Note that some vertices can be labeled by an $r$-tuple accordingly to the
corresponding block.

The following Dyck path corresponds to the word $((abc)d)(ef)(g(hi))$

.

It is important to distinguish $\mathbf{PaT}$ from the actual quasigroup
providing the left and right translation maps.

######  Proposition 3.0.2.

The parenthesised translated words $\mathbf{PaT}$ form a category.

  * •

Objects are the collection $M_{n}(A_{n})$ of parenthesised words of length
$n$, where $n\geq 1$ formed from the alphabet $A_{n}$.

  * •

Morphisms are non-empty for words of the same length. These morphisms are
given by the componentwise action on the $n$-tuple of letters given by the
translation maps $L$ and $R$ applied to the letters of the words. Translation
maps can permute letters of a word or shift a given letter to another one and
obey to the multiplication table given by the Latin square.

###### Proof.

Objects are the collection of parenthesised words of size $n$, where $n\geq
0$, formed from a $Card(n)$ alphabet $A_{n}$ and where letters are allowed to
repeat. Morphisms map an $n$-sized word to another $n$-sized word, using the
left and right translation maps. Those maps act accordingly to the
corresponding $n\times n$ Latin square, on the letters of the word.
Composition of the translation maps are clearly allowed. There exists an
identity morphism $Id$ so that given a word $w$ we have that $Id_{w}:w\to w$.
This is possible since for any $L_{r}(x):x\mapsto rx=c$ (for $x\in A_{n}$ a
letter of $w$) there exists a divisor of $c$ giving back $x$ (by definition of
a quasigroup). The associativity holds. Consider a sequence
$(x_{1},\cdots,x_{n})$. Then one can act on each letter using the left and
right translation maps so that $(x_{1},\cdots,x_{n})$ is mapped to
$(T_{1}(x_{1}),\cdots,T_{n}(x_{n}))$ where $x_{1},\cdots,x_{n}\in A_{n}$.
$T_{i}$ are the translation maps which can be obtained as a composition of
right and left maps acting on each letter independently. The composition of
translation maps is associative. In relation to this it is important to
distinguish the composition of translation maps operation acting on a sequence
of letters and the multiplication operation on the quasigroup. ∎

###  3.1. Parenthesised modified braid groupoid ${\bf mPaB}$

In this section, we rely on the construction presented in [BN98, BdBHR19,
Dri90], in order to show that the same structure applies for the modified
parenthesised braids as for the parenthesised braids.

Roughly speaking by parenthesised braid we mean a braid whose ends (i.e. top
and bottom lines) correspond to parenthesised ordered points along a line. Let
$B$ be such a parenthesised braid with $n$ strands.

In other words, the operad of parenthesised braids ${\bf PaB}$ is the operad
in groupoids defined as follows (see Def. 6.11 [BdBHR19]).

  * •

The operad of objects is the magma operad, i.e. $Ob({\bf
PaB})=M=\\{M(n)\\}_{n\geq 0}$.

  * •

For each $n\geq 0$, the morphisms of the groupoid ${\bf PaB}(n)$ are morphisms
in the (colored) braid groupoid, denoted ${\bf CoB}(n)=\\{{\bf
CoB}(n)\\}_{n\geq 0}$, where $Hom_{{\bf PaB}(n)}(p,q)=Hom_{{\bf
CoB}(n)}(u(p),u(q))$ with $p,q\in\mathbb{S}_{n}$ and the morphisms are braids
associated to the permutation $qp^{-1}$.

For the reader’s convenience we recall the definition of the collection of
groupoids ${\bf CoB}=\\{{\bf CoB}(n)\\}_{n\geq 0}$, following Def. 6.1
[BdBHR19]:

  * •

${\bf CoB}(0)$ is the empty groupoid.

  * •

For $n>0$, the set of objects $Ob({\bf CoB}(n))$ is $\mathbb{S}_{n}$. For our
own purposes, we propose to modify here the classical perspective on this
object by defining the generators of $\mathbb{S}_{n}$ from the point of view
of translation maps, i.e. given by some specific combinations of translation
maps, lying in the space of all translation maps denoted $T_{n}$.

  * •

A morphism in ${\bf CoB}(n)$ from $p$ to $q$ is a braid $\alpha\in B(n)$ whose
associated permutation is $qp^{-1}$.

The categorical composition in ${\bf CoB}(n)$ is given by the concatenation
operation of braids

$Hom_{{\bf CoB}(n)}(p,q)\times Hom_{{\bf CoB}(n)}(q,t)\to Hom_{{\bf
CoB}(n)}(p,t)$

inherited from the braid group. We write $a\cdot b$ for the categorical
composition of $a$ and $b$.

######  Remark 3.1.1.

As one can see, this type of object fits the description of errors of type (1)
discussed earlier. Those errors are mainly given by a permutation of letters
in a word. However, this forms a very restrictive subclass of possible
mistakes. Moreover, it is rare to form sentences of words having all letters
distinct. Therefore, we add the class (2) of possible errors to our
investigations and so it is necessary to modify the definition of ${\bf PaB}$
slightly so as to obtain a larger panel of possible errors.

The above definitions being settled, we introduce the notion of modified
parenthesised braid ${\bf mPaB}$. It is reminiscent to its original version
${\bf PaB}$, in the sense that objects are given by parenthesised words.
Somehow, since letters are allowed to repeat in a word, one needs to introduce
a modification of the braid. This modification of the braid is given by
introducing two supplementary operations: the pinching operation and the
attaching operation. These operations are a geometric representation of the
left (resp. right) translation of a letter into another one, if this letters
has already been used in the word.

######  Definition 2.

Consider a pair of strands in a given braid. We say that there exists a
pinching operation whenever those two strands are pinched together at a point.
This point lies neither on the top nor on the bottom lines of the braid. We
say that there exists an attaching operation if the pinching lies on either
the the top or bottom lines of the braid.

Figure 1. Pinching and attaching points

See an illustration of this in Fig. 1, where a pinching point is presented
between the strands starting at $a$ and $b$ and an attaching point is
presented for the strands starting at $b,c,d$. The attaching operation, occurs
during the transformation of the word $abcd$ into the word $addd$.

######  Remark 3.1.2.

Note that the pinching/ attaching does not imply that the strands have been
intertwined. An intertwining of two strands amounts to solving equation 2 for
a pair of translation maps applied to a pair of letters.

######  Example 3.

We provide an additional example of a pinched (modified) braid in Fig. 2. As
one can see, we have a pair of parenthesised words $(ab)(cd)$ on the top line
and $a(b(cd))$ on the bottom line. The pinching occurs after that $c,d$ and
$a,b$ are swapped and the word morphism maps $(ab)(cd)\to(ba)(dc)$. The
following translation map applied to $c$ (given by $L_{x}(c)=d$) gives the new
word $b(a(dd))$, where two strands are pinched at $d$.

Figure 2. One pinching point

We are interested now in considering the errors of type (4),(5) and (6). This
implies adding letters, losing letters or even duplicating letters. First, we
introduce the modified parenthesised braids.

######  Definition 3.

Let ${\bf mPaB}$ be the category whose objects are parenthesised words and
morphisms are given by a pair
$(P_{\star},\sum_{j=1}^{k}\beta_{j}\tilde{B}_{j})$, constituted from:

  * •

a morphism in the category of parenthesised translations, denoted $P_{\star}$;

  * •

the linear sum of parenthesised modified braids
$\sum_{j=1}^{k}\beta_{j}\tilde{B}_{j}$ is defined such that the skeleton of
$\tilde{B}_{j}$ is $P_{\star}$; the coefficients $\beta_{i}$ lie in some
"ground algebra".

The composition law in ${\bf mPaB}$ is given by the bilinear extension of the
composition law of modified parenthesised braids.

######  Proposition 3.1.1.

The modified parenthesised braids ${\bf mPaB}$ forms a groupoid where:

  * •

The objects are the collections $n$-sized parenthesised translated words.

  * •

For each $n\geq 0$, the morphisms are in the modified braid groupoid
$\bf{mCoB}(n)$, where: $Hom_{{\bf
mPaB}(n)}(p_{\star},q_{\star})=Hom_{\bf{mCoB}(n)}(u(p_{\star}),u(q_{\star})),$
with $p_{\star},q_{\star}\in T_{n}$ being translations.

The symbol $\bf{mCoB}=\\{\bf{mCoB}(n)\\}_{n\geq 0}$ defines the modified
(colored) braids which consists of a collection of groupoids $\bf{mCoB}(n)$
defined as follows.

  * •

$\bf{mCoB}(0)$ is the empty set.

  * •

For $n>0$, the set of objects $Ob(\bf{mCoB}(n))$ are the translations $T_{n}$
(containing $\mathbb{S}_{n}$) on a set of $n$ elements, where rules of
translating elements are given by the corresponding $n\times n$ Latin squares.

  * •

A morphism in $\bf{mCoB}(n)$ from the translation $p_{\star}$ to the
translation $q_{\star}$ is a modified braid $\alpha_{\star}\in mB(n)$ whose
associated translation is $q_{\star}p_{\star}^{-1}$.

###### Proof.

By definition, a groupoid is a small category in which every morphism is an
isomorphism (i.e. it is invertible). A groupoid is given by a set of objects;
here we take the collection of $n$-sized parenthesised words.

For each pair of objects $w$ and $w^{\prime}$ in the set of parenthesised
words, there exists a (possibly empty) set of morphisms from $w$ to
$w^{\prime}$. Here those morphisms are given by translating one $n-$word into
another one by using the translation maps $L$ and $R$. This morphism can
amount to a permutation of the letters of the words (like in the classical
${\bf PaB}$ case) but does not have to. In particular, letters can be shifted
into other letters, creating thus a word where letters repeat.

Now, for every word $w$, there is a designated element $\mathrm{id}_{w}$. This
is due to the fact that in a quasi group every element is invertible and that
we can, in addition, add the notion of neutral element giving the identity
(and forming thus a loop).

For each triple of objects $w,w^{\prime}$, and $w^{\prime\prime}$, one has a
composition of translation maps allowing the morphism $f:w\to w^{\prime}$ to
be composed with $g:w^{\prime}\to w^{\prime\prime}$ and giving $gf:w\to
w^{\prime\prime}$. Furthermore the morphisms are invertible (and this follows
from the definition of translation maps). This construction then leads to the
modified braids. For a given translation $p$ and $q$ of a word, one defines an
associated modified braid, whose translation is $qp^{-1}$. The domain and
range are the parenthesised words corresponding to the permutations $p$ and
$q$ respectively.

∎

In order to construct a “tower” of modified parenthesised braids, the key
setup is already existent for the category ${\bf PaB}$ where one has the
extension operations, cabling operations, strand removal operations. The whole
point of the next proposition and lemma will be to first define rigorously the
modified parenthesised braid groupoid and to show that the extension, cabling
and strand removal operations can be inherited on this new object.

######  Lemma 3.1.1.

Let $({\bf PaB},d_{i},s_{i},d_{0})$ be the category of parenthesised braids,
equipped with the three operations known as: extension operation $d_{0}$,
cabling operation $d_{i}$ and strand removal $s_{i}$ operation. Then, those
operations are inherited on the category ${\bf mPaB}$.

###### Proof.

Consider $B$ a parenthesised braid with $n$ strands.

  * •

Extension operations. Given a braid $B$, one adds on the left-most (or right-
most) side a straight strand, with ends regarded as outer-most. This operation
of adding a straight strand does not encounter any obstruction for the
modified braids and so it is inherited from ${\bf PaB}$.

  * •

Cabling operations. For $1\leq i\leq n$, let us consider the parenthesised
braid obtained from $B$ by doubling its $i$-th strand (counting at the
bottom). This cabling operation can be applied in any situation: either when
strands are separated as in the classical braid setting or when they are
attached/ pinched. So again this operation is inherited from ${\bf PaB}$.

  * •

Strand removal operations. For $1\leq i\leq n$ be the parenthesised braid
obtained from $B$ by removing its $i$-th strand (counting at the bottom).
Removing a strand in the modified braid holds also in this case.

So, all three operations are well defined for ${\bf mPaB}$. ∎

We will now prove that ${\bf PaB}$ is a full subcategory of ${\bf mPaB}$.

######  Corollary 3.1.1.

The category ${\bf PaB}$ is a full subcategory of ${\bf mPaB}$ i.e. there
exists a full inclusion:

${\bf PaB}\hookrightarrow{\bf mPaB}.$

###### Proof.

To have a full subcategory of ${\bf mPaB}$ it is necessary that for any
objects $x,y$ in ${\bf PaB}$ every morphism of in ${\bf mPaB}$ is also in
${\bf PaB}$. This is a true statement since any permutation is given by
translations maps $L$ and $R$ (satisfying conditions (2) for any
transposition). ∎

In the setting of the category ${\bf PaB}$, there exists a functor
${\mathbf{S}}$ called the skeleton functor on whose image is the category of
parenthesised permutations $\mathbf{PaP}$. The operations $d_{i},s_{i}$ of
cabling and strand removal operations are naturally defined on $PaP$. The
skeleton functor ${\mathbf{S}}$ intertwines the $d_{i}$’s and the $s_{i}$’s
acting on parenthesised braids and on parentesized permutations. The same type
of object exists for ${\bf mPaB}$ and parenthesised translations.

Denote the parenthesised translations $\mathbf{PaT}$. The skeleton functor
${\mathbf{S}}_{\mathbf{PaT}}$ for the ${\bf mPaB}$ is the identity on objects,
where objects are parenthesised words in $\mathbf{PaT}$ i.e words equipped
with parenthesis and where letters can be permuted or translated, giving thus
possibly parenthesised words with non distinct letters.

######  Proposition 3.1.2.

The category $\mathbf{PaT}$ together with the functor
${\mathbf{S}}_{\mathbf{PaT}}:{\bf mPaB}\to\mathbf{PaT}$ is a fibered linear
category.

###### Proof.

The category $\mathbf{PaT}$ together with the functor
${\mathbf{S}}_{\mathbf{PaT}}:{\bf mPaB}\to\mathbf{PaT}$ forms a fibered linear
category, for the following reasons. First, $\mathbf{PaT}$ has the same
objects as ${\bf mPaB}$ and the skeleton functor is the identity on objects.
Secondly, the inverse image ${\mathbf{S}}_{\mathbf{PaT}}^{-1}(P_{\star})$ of
every morphism $P_{\star}\in\mathbf{PaT}$ is a linear composition of left and
right translation maps $L$ (and $R$) and, similarly to the case of the
parenthesised braids, it forms a linear space. The composition maps in ${\bf
mPaB}$ are also bilinear in the natural sense. ∎

###  3.2. The Grothendieck–Teichmüller group and modified parenthesised braids

We now discuss the following theorem.

######  Theorem 3.2.1.

The pro-unipotent Grothendieck–Teichmüller group is contained in the groups of
structure preserving automorphisms $Aut(\widehat{{\bf mPaB}})$.

In order to prove this statement, we apply the method on fibered linear
categories such as shown in detail in Sec.2.1.1. [BN98]). Consider ${\bf
mPaB}$ and $\mathbf{PaT}$, being respectively the categories of modified
parenthesised braids and parenthesised translations. Define a subcategory of
the fibered linear category $({\bf mPaB},{\mathbf{S}}:{\bf
mPaB}\to\mathbf{PaT})$ as follows. Let $P_{\star}$ be a morphism in
$\mathbf{PaT}$. Choose a linear subspace in each
${\mathbf{S}}^{-1}(P_{\star})$, so that the system of subspaces chosen is
closed under composition. Closed under composition means that two modified
braids (such that the bottom line of the first modified braid coincides with
the top line of the second one) lie both in the linear subspace generated by
${\mathbf{S}}^{-1}(P_{\star})$ and defines another modified braid belonging to
${\mathbf{S}}^{-1}(P_{\star})$. We construct an ideal ${\mathbf{I}}$ in $({\bf
mPaB},{\mathbf{S}}:{\bf mPaB}\to\mathbf{PaT})$, which is a subcategory. The
quotient ${\bf mPaB}/{\mathbf{I}}$ of the fibered linear category ${\bf mPaB}$
by the ideal ${\mathbf{I}}$ is again a fibered linear category.

These fibered linear categories are compatible with further operations
allowing the construction of the inverse limit of an inverse system of fibered
linear categories (fibered in a compatible way over the same category of
skeletons). So, if ${\mathbf{I}}$ is an ideal in a fibered linear category
${\mathbf{B}}$, one can form the ${\mathbf{I}}$-adic completion and this
${\mathbf{I}}$-adic completion is again a filtered fibered linear category.

######  Lemma 3.2.1.

There exists a tower of modified parenthesised braids $(\widehat{{\bf
mPaB}},\widehat{{\bf mPaB}}\to\mathbf{PaT},d_{i},s_{i})$, where $\widehat{{\bf
mPaB}}$ is the unipotent completion of ${\bf mPaB}$.

###### Proof.

Define the subcategory of the fibered linear category $({\bf
mPaB},{\mathbf{S}}_{\mathbf{PaT}}:{\bf mPaB}\to\mathbf{PaT})$ as follows. Let
$P_{\star}$ be a morphism in $\mathbf{PaT}$. We choose a linear subspace in
each ${\mathbf{S}}^{-1}(P_{\star})$, so that the system of subspaces chosen is
closed under composition (two translations such that the range of the first
translation $T_{1}$ is the domain of the second $T_{2}$ and both lying in the
linear subspace in ${\mathbf{S}}^{-1}(P_{\star})$ define a translation
$T_{1}\circ T_{2}$ also in ${\mathbf{S}}^{-1}(P_{\star})$).

As mentioned earlier, an ideal in $({\bf
mPaB},{\mathbf{S}}_{\mathbf{PaT}}:{\bf mPaB}\to\mathbf{PaT})$ is a subcategory
${\mathbf{I}}$ so that if at least one of the two composable morphisms $T_{1}$
and $T_{2}$ in $\mathbf{PaT}$ is actually in ${\mathbf{I}}$, then their
composition $T_{1}\circ T_{2}$ is also in ${\mathbf{I}}$. The ideal
${\mathbf{I}}^{m}$ is such that morphisms of ${\mathbf{I}}^{m}$ are all those
morphisms in ${\bf mPaB}$ that can be presented as compositions of $m$
morphisms in ${\mathbf{I}}$.

In particular, given that ${\mathbf{I}}$ is an ideal of a fibered linear
category ${\bf mPaB}$, one can form the ${\mathbf{I}}$-adic completion
$\widehat{{\bf mPaB}}=\lim_{m\to\infty}{\bf mPaB}/{\mathbf{I}}^{m}$, where the
${\mathbf{I}}$-adic completion is a filtered fibered linear category.

Take ${\mathbf{I}}$ to be the augmentation ideal of ${\bf mPaB}$ formed from
all pairs $(P,\beta_{j}T_{j})$ in which $\sum\beta_{j}=0$. Powers of this
ideal defines the unipotent filtration of ${\bf mPaB}$, which is denoted
$\mathcal{F}_{\bf mPaB}={\mathbf{I}}^{m+1}$.

Let ${\bf mPaB}^{(m)}={\bf mPaB}/\mathcal{F}_{m}{\bf mPaB}={\bf
mPaB}/{\mathbf{I}}^{m+1}$ be the $m$-th unipotent quotient of ${\bf mPaB}$,
and let $\widehat{{\bf mPaB}}=\lim_{m\to\infty}{\bf mPaB}^{(m)}$, be the
unipotent completion of ${\bf mPaB}$. The fibered linear categories inherit
the operations $d_{i}$ and $s_{i}$ and a coproduct and filtration
$\mathcal{F}_{*}$. ∎

Finally this construction leads to considering the automorphism group of the
tower of modified braids $Aut(\widehat{{\bf mPaB}})$, being the group of all
functors $\widehat{{\bf mPaB}}\to\widehat{{\bf mPaB}}$, covering the skeleton
functor, intertwining $d_{i},s_{i},\square$ (the coproduct functor
$\square:{\bf mPaB}\to{\bf mPaB}\otimes{\bf mPaB}$) and fixing the elementary
braid $\sigma$ (a crossing of two strands).

###### Proof of Theorem 3.2.1.

We have shown that $\widehat{{\bf mPaB}}$ is an enriched version of the
construction of $\widehat{PaB}$ in [BN98] (in the sense that it inherits its
properties and operations but has some additional structures). As well we
obtained that ${\bf PaB}$ is a subcategory of ${\bf mPaB}$.

By definition, we have that $\widehat{GT}=Aut(\widehat{{\bf PaB}})$, where
$Aut(\widehat{{\bf PaB}})$ is the group of all functors $\widehat{{\bf
PaB}}\to\widehat{{\bf PaB}}$ that cover the skeleton functor, intertwine
$d_{i},s_{i}$ and $\square$ and fixes $\sigma$. The inclusion of ${\bf PaB}$
in ${\bf mPaB}$ implies that $Aut(\widehat{{\bf PaB}})$ is included in
$Aut(\widehat{{\bf mPaB}})$. So, the pro-unipotent Grothendieck–Teichmüller
group is contained in the groups of structure preserving automorphisms
$Aut(\widehat{{\bf mPaB}})$. ∎

Finally, using the inclusion theorem of [Bro12] stating that the motivic
Galois group is included in $Aut(\widehat{{\bf PaB}})$, we can conclude that:

######  Corollary 3.2.1.

The motivic Galois group is contained in the automorphism group
$Aut(\widehat{{\bf mPaB}})$.

We can interpret $\widehat{{\bf mPaB}}$ as modelling a situation where one
considers infinitely many errors occurring. The information that it tells us
is that the in a pro-unipotent completion of the space of ways of making
errors has among others the behaviour of the motivic Galois group encapsulated
within it.

###  3.3. Conjectures and open questions

Moufang loops turn out to be central in geometry of information, in particular
for statistical manifolds (related to exponential families) and codes/error-
codes. For instance, symmetries of spaces of probability distributions,
endowed with their canonical Riemannian metric of information geometry, have
the structure of a commutative Moufang loop.

In a different setting, there exists a connection between Moufang loops
algebraic geometry. Recall from [Man86] the relation between Moufang loops and
the set of algebraic points of a smooth cubic curve in a projective plane
$\mathbb{P}^{2}_{K}$ over a field $K$. The set $E$ of $K$-points of such a
curve $X$ forms a $CML$ with composition law $x\circ y=u\star(x\star y)$, if
$u+x+y$ is the intersection cycle of $X$ with a projective line
$\mathbb{P}^{1}_{K}\subset\mathbb{P}^{2}_{K}$.

Given that symmetries of statistical manifolds have the structure of $CML$ and
that Manin’s conjecture (coming from algebraic geometry) is true in the case
of statistical manifolds, it leads to think that there is a stronger
connection between the $CML$ coming from algebraic geometry and the $CML$ in
the statistical manifolds. So, an intriguing question following from the
properties of statistical manifolds defined above would be to determine
whether the set $E$ of $K$-points of a pre-Frobenius statistical manifold has
the structure of a $CML$.

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