# Discovering Unsupervised Behaviours from Full-State Trajectories

Luca Grillotti & Antoine Cully
Imperial College London
London, UK
{luca.grillotti16<EMAIL_ADDRESS>

###### Abstract

Improving open-ended learning capabilities is a promising approach to enable
robots to face the unbounded complexity of the real-world. Among existing
methods, the ability of Quality-Diversity algorithms to generate large
collections of diverse and high-performing skills is instrumental in this
context. However, most of those algorithms rely on a hand-coded behavioural
descriptor to characterise the diversity, hence requiring prior knowledge
about the considered tasks. In this work, we propose an additional analysis of
Autonomous Robots Realising their Abilities; a Quality-Diversity algorithm
that autonomously finds behavioural characterisations. We evaluate this
approach on a simulated robotic environment, where the robot has to
autonomously discover its abilities from its full-state trajectories. All
algorithms were applied to three tasks: navigation, moving forward with a high
velocity, and performing half-rolls. The experimental results show that the
algorithm under study discovers autonomously collections of solutions that are
diverse with respect to all tasks. More specifically, the analysed approach
autonomously finds policies that make the robot move to diverse positions, but
also utilise its legs in diverse ways, and even perform half-rolls.

##  1 Introduction

One of the motivations of using learning algorithms in robotics is to enable
robots to discover on their own their abilities. Ideally, a robot should be
able to discover on its own how to manipulate new objects (Johns et al., 2016)
or how to adapt its behaviours when facing an unseen situation like a
mechanical damage (Cully et al., 2015). However, despite many impressive
breakthroughs in learning algorithms for robotics applications during the last
decade (Akkaya et al., 2019; Hwangbo et al., 2019), these methods still
require a significant amount of engineering to become effective, for instance,
to define reward functions, state definitions, or characterise the expected
behaviours.

Finding all possible behaviours for a robot is an open-ended process, as the
set of all achievable robot behaviours is unboundedly complex. There are
plenty of different ways to execute each type of behaviour. For instance, for
moving forward, rotating or jumping, the robot can move at different speeds,
and use different locomotion gaits. Supplementary behaviours can emerge from
interactions with objects and with other agents.

An attempt to discover autonomously the behaviours of a robot has been
proposed with the AURORA algorithm (Cully, 2019; Grillotti & Cully, 2021).
This algorithm leverages the creativity of Quality-Diversity (QD) optimisation
algorithms to generate a collection of diverse and high-performing behaviours,
called a behavioural repertoire. QD algorithms usually require the definition
of a manually-defined Behavioural Descriptor (BD) to characterise the
different types of behaviours contained in the repertoire. Instead, AURORA
uses dimensionality-reduction techniques to perform the behavioural
characterisation in an unsupervised manner. The resulting algorithms enable
robots to autonomously discover a large diversity of behaviours without
requiring a user to define a performance function or a behavioural descriptor.
Nevertheless, in prior work, AURORA has only been used to learn behavioural
characterisations from hand-defined sensory data. For example, Grillotti &
Cully (2021) use as sensory data a robot picture taken at the end of an
episode from an appropriate camera angle. In that case, a consequent amount of
prior information is provided to the algorithm: only the end of the episode
should be considered (instead of the full trajectory), and only the given
camera angles should be used (instead of other camera angles or joints
positions).

In this work, we propose an extended analysis of AURORA, where behavioural
characterisations are learned based on the complete trajectory of the robot
raw states. In that sense, no prior information is injected in the sensory
data. We evaluated AURORA on a simulated hexapod robot with a neural network
controller on three tasks: navigation, moving forward with a high velocity,
and performing half-rolls. Furthermore, we compare AURORA to several
baselines, all of which use hand-coded BDs. In most cases, the collections
obtained via AURORA present more diversity than those obtained from hand-coded
QD algorithms. In particular, AURORA automatically discovers behaviours that
make the robot navigate to diverse positions, while using its legs in diverse
ways, and perform half-rolls.

##  2 Background

###  2.1 Quality-Diversity Algorithms

Quality-Diversity (QD) algorithms are a subclass of evolutionary algorithms
that aim at finding a container of both diverse and high-performing policies.
In addition to standard evolutionary algorithms, QD algorithms consider a
Behavioural Descriptor (BD), which is a low-dimensional vector that
characterises the behaviour of a policy. QD algorithms use the BDs to quantify
the novelty of a policy with respect to the solutions already in the
container. The container of policies is filled in an iterative manner,
following these steps: (1) policies are selected from the container and
undergo random variations (e.g. mutations, cross-overs); (2) their performance
score and BD are evaluated; and (3) we try to add them back to the container.
If they are novel enough compared to solutions that are already in the
container, they are added to the container. If they are better than similar
policies in the container, they replace these policies.

###  2.2 Autonomous Robots Realising their Abilities (AURORA)

Quality diversity algorithms are a promising tool to generate a large
diversity of behaviours in robotics. However, the definition of the BD space
might not always be straightforward when the robot or its capabilities are
unknown. AURORA is a QD algorithm designed to discover the abilities of
robots, by maximising the diversity of behaviours contained in the repertoire
in an unsupervised manner. AURORA automatically defines the BD by encoding the
high-dimensional sensory data collected by the robot during an episode. The
encoding can be achieved using any dimensionality reduction technique, such
as, Auto-Encoder (AE) (Grillotti & Cully, 2021) or Principal Component
Analysis (Cully, 2019).

To generate a behavioural repertoire, AURORA alternates between QD phases and
Encoder phases. During the QD phase, policies undergo the same process as in
standard QD algorithms: selection from the container, evaluation, and attempt
to add to the container. However, the evaluation is performed in a slightly
different way: instead of a low-dimensional BD the QD task returns high-
dimensional sensory data, that is then encoded using the dimensionality
reduction technique. The latent encoding of the sensory data is used as the BD
for the rest of the QD phrase.

During the Encoder phase, the dimensionality reduction structure (e.g. the AE)
is trained using the high-dimensional data from all the policies present in
the container. Once the encoder is trained, the unsupervised BDs are
recomputed with the new encoder for all policies present in the container.

Alternating these two phases enables AURORA to progressively build its
behavioural repertoires by discovering new solutions, while refining its
encoding of the high-dimensional sensory data generated by the robot every
time new solutions are added to the archive. In this work, the sensory data
collected by the robot corresponds to the full-state trajectory of the robot.

##  3 Learning Diverse Behaviours from Full-State Trajectories

In this section, we provide a theoretical analysis of the AURORA algorithm, to
support its ability to generate a large collection of diverse behaviours in an
unsupervised manner. In essence, we aim to find a container of policies
exhibiting diverse behaviours. We consider that a behaviour is defined as the
full-state trajectory: ${\bm{{s}}}_{1:T}$. Then the overall problem of finding
diverse behaviours $\left({\bm{{s}}}_{1:T}^{(i)}\right)_{i}$ can be expressed
as an entropy maximisation problem. Considering the behaviour
${\bm{{s}}}_{1:T}$ as a random variable, we aim to maximise the entropy of
${\bm{{s}}}_{1:T}$, written ${H\left({{{\bm{{s}}}_{1:T}}}\right)}$. In the
following sections we will explain (1) why it can be ineffective to maximise
directly the diversity of full-state trajectories ${\bm{{s}}}_{1:T}$; and (2)
how AURORA can be used to maximise indirectly this diversity.

###  3.1 Problem with High-dimensional Behavioural Descriptors

Maximising the entropy of full-state trajectories
${H\left({{{\bm{{s}}}_{1:T}}}\right)}$ is theoretically possible by using a QD
algorithm with ${\bm{{s}}}_{1:T}$ as a Behavioural Descriptor (BD). However,
it is computationally inefficient to use QD algorithms with high-dimensional
BDs.

Indeed, as the dimensionality of that problem is high, a consequent number of
samples of ${\bm{{s}}}_{1:T}$ is required to optimise
${H\left({{{\bm{{s}}}_{1:T}}}\right)}$. So a significant number of policies is
needed in the container. However, QD algorithms become less efficient when the
number of policies increases. When the number of policies in the container
increases, the selection pressure of the container decreases, which means that
the policies from the container are mostly less likely to be selected (Cully &
Demiris, 2018).

Also, in QD algorithms, unstructured containers always rely on the k-nearest
neighbours algorithm to compute the novelty of a policy. And the complexity of
that algorithm can be
$O\left(\dim{\mathcal{B}}\log\lvert\mathcal{C}\rvert\right)$ in average
(Bentley, 1975), and $O\left(\dim{\mathcal{B}}\lvert\mathcal{C}\rvert\right)$
in the worst case ($\mathcal{B}$ refers to the BD space, and
$\lvert\mathcal{C}\rvert$ is the number of policies in the container
$\mathcal{C}$). Hence, the computational complexity of one QD generation
increases linearly with dimensionality of the BD $\dim{\mathcal{B}}$. That is
why the overall procedure of adding policies to the container at each QD
generation may become too computationally expensive if the dimension of
$\mathcal{B}$ increases significantly.

###  3.2 Learning Low-dimensional Descriptors from High-dimensional State
Trajectories

The encoder update phase of AURORA optimises the parameters
${\bm{{\eta}}}_{\mathcal{E}}$ of the encoder $\mathcal{E}$; this encoder is
then used to define a low-dimensional BD ${\bm{{b}}}_{\mathcal{E}}$ from state
trajectories: ${\bm{{b}}}_{\mathcal{E}}=\mathcal{E}({\bm{{s}}}_{1:T})$. Figure
1 summarises the dependencies between the different variables considered in
standard QD algorithms and AURORA. The mutual information
$I\left(\cdot,\cdot\right)$ between ${\bm{{s}}}_{1:T}$ and
${\bm{{b}}}_{\mathcal{E}}$ can be expressed in two symmetrical ways:

$\displaystyle I\left({\bm{{s}}}_{1:T},{\bm{{b}}}_{\mathcal{E}}\right)$
$\displaystyle={H\left({{{\bm{{s}}}_{1:T}}}\right)}-H\left({{\bm{{s}}}_{1:T}}\mid{{\bm{{b}}}_{\mathcal{E}}}\right)$
(1) $\displaystyle I\left({\bm{{s}}}_{1:T},{\bm{{b}}}_{\mathcal{E}}\right)$
$\displaystyle={H\left({{{\bm{{b}}}_{\mathcal{E}}}}\right)}-H\left({{\bm{{b}}}_{\mathcal{E}}}\mid{{\bm{{s}}}_{1:T}}\right)$
(2)

As ${\bm{{b}}}_{\mathcal{E}}$ is a function of ${\bm{{s}}}_{1:T}$, we have:
$H\left({{\bm{{b}}}_{\mathcal{E}}}\mid{{\bm{{s}}}_{1:T}}\right)=0$. Then, from
equations 1 and 2 we get:

$\displaystyle{H\left({{{\bm{{s}}}_{1:T}}}\right)}$
$\displaystyle={H\left({{{\bm{{b}}}_{\mathcal{E}}}}\right)}+H\left({{\bm{{s}}}_{1:T}}\mid{{\bm{{b}}}_{\mathcal{E}}}\right)$
(3)
$\displaystyle={H\left({{{\bm{{b}}}_{\mathcal{E}}}}\right)}+H\left({{\bm{{s}}}_{1:T}}\mid{\mathcal{E}\left({\bm{{s}}}_{1:T}\right)}\right)$
(4)

According to equation 4, ${H\left({{{\bm{{s}}}_{1:T}}}\right)}$ can be
decomposed into two components:

  * •

The entropy on the BD space ${H\left({{{\bm{{b}}}_{\mathcal{E}}}}\right)}$,
which can be maximised via a QD algorithm.

  * •

The conditional entropy of the state trajectories with respect to their
encoding
$H\left({{\bm{{s}}}_{1:T}}\mid{\mathcal{E}\left({\bm{{s}}}_{1:T}\right)}\right)$.
This term quantifies the information that is lost between ${\bm{{s}}}_{1:T}$
and its encoding
${\bm{{b}}}_{\mathcal{E}}=\mathcal{E}\left({\bm{{s}}}_{1:T}\right)$.

Inequality 4 gives a lower bound on the entropy of state trajectories:

$\displaystyle{H\left({{{\bm{{s}}}_{1:T}}}\right)}\geq{H\left({{{\bm{{b}}}_{\mathcal{E}}}}\right)}$
(5)

And Fano’s inequality (Cover & Thomas, 2012, p. 38) gives a bounding on the
differences between those entropies:

$\displaystyle{H\left({{{\bm{{s}}}_{1:T}}}\right)}-{H\left({{{\bm{{b}}}_{\mathcal{E}}}}\right)}$
$\displaystyle=H\left({{\bm{{s}}}_{1:T}}\mid{\mathcal{E}\left({\bm{{s}}}_{1:T}\right)}\right)$
(6)
$\displaystyle{H\left({{{\bm{{s}}}_{1:T}}}\right)}-{H\left({{{\bm{{b}}}_{\mathcal{E}}}}\right)}$
$\displaystyle\leq{H\left({{e}}\right)}+P(e)\log\lvert\mathcal{X}_{\mathcal{O}^{T}}\rvert$
(7)

where $P(e)=P({\bm{{s}}}_{1:T}\neq\widehat{{\bm{{s}}}_{1:T}})$,
$\widehat{{\bm{{s}}}_{1:T}}$ represents a reconstruction of the state
trajectories ${\bm{{s}}}_{1:T}$ from the encoding
$\mathcal{E}({\bm{{s}}}_{1:T})={\bm{{b}}}_{\mathcal{E}}$, and
$\mathcal{X}_{\mathcal{O}^{T}}$ represents the discretised support of the
state trajectories ${\bm{{s}}}_{1:T}$.

As explained in section 2.2, AURORA alternates between two phases: an encoder
update phase, and a QD phase. During the encoder update phase, we minimise the
difference
${H\left({{{\bm{{s}}}_{1:T}}}\right)}-{H\left({{{\bm{{b}}}_{\mathcal{E}}}}\right)}={H\left({{{\bm{{s}}}_{1:T}}}\right)}-{H\left({{\mathcal{E}({\bm{{s}}}_{1:T})}}\right)}$
by training the encoder $\mathcal{E}$. Indeed, training $\mathcal{E}$ will
reduce the probability of reconstruction error $P(e)$, with the intention of
lessening the difference
${H\left({{{\bm{{s}}}_{1:T}}}\right)}-{H\left({{{\bm{{b}}}_{\mathcal{E}}}}\right)}$
(see inequality 7). During the QD phase, now that the difference between both
sides of inequality 5 is minimised, we increase the lower bound of
${H\left({{{\bm{{s}}}_{1:T}}}\right)}$, namely
${H\left({{{\bm{{b}}}_{\mathcal{E}}}}\right)}$. As ${\bm{{b}}}_{\mathcal{E}}$
is a low-dimensional BD, maximising
${H\left({{{\bm{{b}}}_{\mathcal{E}}}}\right)}$ can be intuitively achieved by
applying a standard Quality-Diversity algorithm, with
${\bm{{b}}}_{\mathcal{E}}$ as BD.

In the end, the two phases of AURORA successively (1) train the encoder to
minimise the difference between the entropy of successive states
${H\left({{{\bm{{s}}}_{1:T}}}\right)}$ and its lower bound
${H\left({{\mathcal{E}({\bm{{s}}}_{1:T})}}\right)}$, and (2) use QD
iterations, to maximise the lower bound
${H\left({{{\bm{{b}}}_{\mathcal{E}}}}\right)}={H\left({{\mathcal{E}({\bm{{s}}}_{1:T})}}\right)}$.
This intuitively explains why AURORA can be used to maximise the diversity of
full-state trajectories ${\bm{{s}}}_{1:T}$.

${\bm{{\theta}}}$${\bm{{s}}}_{1}$${\bm{{s}}}_{2}$$\ldots$${\bm{{s}}}_{T}$${\bm{{b}}}_{\mathcal{T}}$
(a) Graphical Model of standard Quality-Diversity optimisation tasks

${\bm{{\theta}}}$${\bm{{s}}}_{1}$${\bm{{s}}}_{2}$$\ldots$${\bm{{s}}}_{T}$${\bm{{b}}}_{\mathcal{E}}$${\bm{{\eta}}}_{\mathcal{E}}{}$
(b) Graphical Model of an optimisation task using AURORA

Figure 1:  Graphical Models of the Quality-Diversity optimisation algorithms
in the standard case (1(a)) and using AURORA (1(b)). ${\bm{{\theta}}}$
corresponds to the parameters of the agent’s policy. At each time-step $t$,
${\bm{{s}}}_{t}$ represent the state of the agent. ${\bm{{b}}}_{\mathcal{T}}$
represents the task behavioural descriptor based on the states of the robot;
${\bm{{b}}}_{\mathcal{E}}$ is a behavioural descriptor based on the successive
states of the robot ${\bm{{s}}}_{1:T}$. Finally,
${\bm{{\eta}}}_{\mathcal{E}}{}$ corresponds to the parameters of the encoder
$\mathcal{E}$ used to reduce the dimensionality of the state trajectories
${\bm{{s}}}_{1:T}$.

##  4 Related Works

The problem of autonomously finding diverse behaviours has been addressed with
Mutual-Information (MI) maximisation approaches (Gregor et al., 2016;
Eysenbach et al., 2018; Sharma et al., 2019). Such methods aim to maximise the
mutual information between the states explored by the robot, and a latent
variable representing the behaviour. Those MI-based approaches then return a
single policy, which is conditioned on the behaviour latent variable. That
differs from QD algorithms, which return a container of different policies.
Also, MI-maximisation algorithms do not necessarily require the definition of
a BD. For example, in the work of Eysenbach et al. (2018), diverse robotic
behaviours are learned from full robot states. Similarly to AURORA, when the
robot states are described with high-dimensional images, an encoder can be
used to compute tractable low-dimensional representations (Liu & Abbeel,
2021b; a). While sharing the same purpose, QD and MI-maximisation algorithms
are fundamentally different in their core design. Those differences make it
challenging to compare the two approaches in a fair manner.

In QD algorithms, selecting the most appropriate definitions for the BD and
the performance requires a certain level of expertise. For instance, the MAP-
Elites algorithm (Mouret & Clune, 2015; Cully et al., 2015) is often used with
the same hexapod robot, but with two different sets of definitions: 1) the leg
duty-factor for the BD, defined as the time proportion each leg is in contact
with the ground, with the average speed for the performance (Cully et al.,
2015; Cully & Demiris, 2018) or 2) the final location of the robot after
walking during three seconds for the BD, with an orientation error for the
performance (Cully & Mouret, 2013; Duarte et al., 2016; Chatzilygeroudis et
al., 2018). The resulting behavioural repertoires will find different types of
applications. The first set of definitions is designed for fast damage
recovery as it provides a diversity of ways to achieve high-speed gaits, while
the second one is designed for navigation tasks as the repertoire enables the
robot to move in various directions.

To avoid the expertise requirements in the behaviour descriptor definition,
several methods have been proposed to enable the automatic definition of the
behavioural descriptors. As described previously, AURORA aims to tackle this
problem by using a dimensionality reduction algorithm (a Deep Auto-Encoder
(Masci et al., 2011)) to project the features of the solutions into a latent
space and use this latent space as a behavioural descriptor space. TAXONS
(Paolo et al., 2020) is another QD algorithm adopting the same principle; it
demonstrated that this approach can scale to camera images to produce large
behavioural repertoires for different types of robots. The same concept was
also used in the context of Novelty Search prior to AURORA and TAXONS to
generate assets for video games with the DeLeNoX algorithm (Liapis et al.,
2013). All these methods aim to maximise the diversity of the produced
solutions, without specifically considering a downstream task.

A different direction to automatically define a behavioural descriptor is to
use Meta-Learning. The idea is to find the BD definition that maximises the
performance of the resulting collection of solutions when used in a downstream
task. For instance, Bossens et al. (2020) considers the damage recovery
capabilities provided by a behavioural repertoire as a Meta-Learning objective
and search for the linear combination of a pre-defined set of behaviour
descriptors that will maximise this objective. A related approach proposed by
Meyerson et al. (2016) uses a set of training tasks to learn what would be the
most appropriate BD definition to solve a ”test task”.

##  5 Experimental Setup

###  5.1 Agent: Neural-network controlled hexapod

In all our experiments, our agent consists of a simulated hexapod robot
controlled via a neural network controller. The hexapod robot presents 18
controllable joints (3 per leg). Each leg $l$ has two Degrees of Freedom: its
hip angle $\alpha_{l}$ and its knee angle $\beta_{l}$. The ankle angle is
always set to the opposite of the knee angle.

The neural network controller is a Multi-Layer Perceptron with a single hidden
layer of size 8. It takes as input the current joint angles and velocities
$(\alpha_{l},\beta_{l},\dot{\alpha}_{l},\dot{\beta}_{l})_{1\leq l\leq 6}$ (of
dimension 24), and outputs the target values to reach for the joint angles
$(\alpha_{l},\beta_{l})_{1\leq l\leq 6}$ (of dimension 12). The controller has
in total $(24+1)*8+(8+1)*12=308$ independent parameters, operates at a
frequency of $50$Hz, while each episode lasts for 3 seconds. The environment
and the controller are deterministic.

###  5.2 Dimensionality Reduction Algorithm of AURORA

For each evaluated policy, the state trajectory corresponds to the joint
positions, and torso positions and orientations collected at a frequency of
10Hz. In the end, this represents 18 streams
($(\alpha_{i},\beta_{i})_{i=1\ldots 6}$, and the positional and rotational
coordinates of the torso), with each of those streams containing 30
measurements. The dimensionality reduction algorithm used in AURORA is a
reconstruction-based auto-encoder. The input data is using two 1D
convolutional layers, with 128 filters and a kernel of size 3. Those
convolutional layers are followed by one fully-connected linear layer of size
256. The decoder is made of a fully-connected linear layer of size 256,
followed by three 1D deconvolutions with respectively 128, 128 and 18 filters.

The loss function used to train the Encoder corresponds to the mean squared
error between the observation passed as input and the reconstruction obtained
as output. The training is performed using the Adam gradient-descent optimiser
(Kingma & Ba, 2014) with $\beta_{1}=0.9$ and $\beta_{2}=0.999$. As the encoder
needs to be updated less and less frequently as the number of iterations
increase, the number of iterations between two encoder updates is linearly
increased over time: if the first encoder update happens at iteration 10, then
the following encoder updates occur at iterations 30, 60, 100…

###  5.3 QD Tasks

For the hexapod environment, we consider different types of QD Tasks. A QD
task evaluates the diversity and the performance of policies present in the
container returned by a QD algorithm. Each QD task is characterised by a
performance score function ${\bm{{\theta}}}\mapsto f({\bm{{\theta}}})$, and a
BD function ${\bm{{\theta}}}\mapsto{\bm{{b}}}_{\mathcal{T}}({\bm{{\theta}}})$
(where ${\bm{{\theta}}}$ represents the policy parameters).

We consider three QD tasks with the hexapod agent: Navigation (Nav), Moving
Forward (Forw), and Half-roll (Roll).

####  5.3.1 Navigation (Nav)

The Navigation task is inspired from Chatzilygeroudis et al. (2018) and
Kaushik et al. (2020). This task aims at finding a container of controllers
reaching diverse final $(x_{T},y_{T})$ positions in $T=3$ seconds following
circular trajectories. Hence the hand-coded BD in this case is
${\bm{{b}}}_{\mathcal{T}}=(x_{T},y_{T})$, while the performance $f$ is the
final orientation error as defined in the literature (Cully & Mouret, 2013;
Chatzilygeroudis et al., 2018).

####  5.3.2 Moving Forward (Forw)

The Moving Forward task is inspired from the work of Cully et al. (2015),
which aims to obtain a container with a variety of ways to move forward at a
high velocity. The hand-coded BD associated to that task
${\bm{{b}}}_{\mathcal{T}}$ is the Duty Factor, which evaluates the proportion
of time the each leg is in contact with the ground. The performance score
promotes solutions achieving the highest $x$ position at the end of the
episode: $f=x_{T}$.

####  5.3.3 Half-roll (Roll)

Half-roll QD task aims at finding a diversity of ways to perform half-rolls,
i.e. behaviours such that the hexapod ends with its back on the floor (where
the pitch angle of the torso is approximately equal to $-\frac{\pi}{2}$). In
particular, the behaviours are characterised via the final yaw and roll angle
of the torso. And the performance is measured as the distance between the
final pitch angle $\alpha^{pitch}_{T}$ of the hexapod and $-\frac{\pi}{2}$.

${\bm{{b}}}_{\mathcal{T}}=\left(\alpha^{yaw}_{T},\alpha^{roll}_{T}\right)\quad
f=-\left\lvert\alpha^{pitch}_{T}-\left(-\frac{\pi}{2}\right)\right\rvert$ (8)

###  5.4 Metric: Coverage per Minimum Performance

The containers returned by any QD algorithm always result in a trade-off
between the diversity and the total performance of the solutions from the
container. To evaluate the coverage depending on the quality of the solutions,
we study the evolution of the coverage given several minimum performance
scores. With a performance $f_{min}$ and a BD space discretised into a grid,
we define the coverage given minimum performance $f_{min}$ as: the number of
cells filled with policies whose performance is higher than $f_{min}$.

The coverage is calculated by discretising the task Behavioural Descriptor
(BD) space $\mathcal{B}_{\mathcal{T}}$ into a $50\times 50$ grid. The coverage
then corresponds the number of grid cells containing at least one policy from
the unstructured container.

###  5.5 Compared algorithms and variants

We compare AURORA (see Section 2.2), to two different variants having hand-
defined BDs: Hand-Coded-$x$ (HC-$x$) and Mean Streams (MeS).

Hand-Coded-$x$ (HC-$x$): considers a low-dimensional BD that is defined by
hand as the BD of the QD task $x$, where $x$ can correspond to any QD task
(Navigation, Moving Forward or Half-roll). The HC variant corresponds to a
standard QD algorithm with an unstructured archive (Cully & Demiris, 2018) and
the mechanism of container size control introduced in previous work (Grillotti
& Cully, 2021).

Mean Streams (MeS): uses a BD that encompasses more information from the full-
state trajectory than the Hand-Coded variant detailed above. The BD used by
this variant considers the 18 data streams collected by the hexapod, and
average each one of those streams to obtain a BD of dimension 18. The
comparison between this variant and AURORA will be appropriate to evaluate the
utility of the automatic dimensionality reduction algorithm.

###  5.6 Implementation Details

All our experiments were run for 15,000 iterations with a uniform QD selector.
We only use polynomial mutations (Deb & Agrawal, 1999) as variation operators
with $\eta_{m}=10$, and a mutation rate of $0.3$. The target container size
set for all algorithms is set to $N_{\mathcal{C}}^{target}=5{,}000$ for the
Moving Forward and the Half-roll tasks. In the case of the Navigation Task, we
set this number to $N_{\mathcal{C}}^{target}=1{,}500$ to obtain appropriate
comparisons between the different approaches. To keep the container size
around $N_{\mathcal{C}}^{target}$, we perform a container update every
$T_{\mathcal{C}}=10$ iterations (as explained in Section 2.1).

All our implementation is based on Sferes${}_{\text{v2}}$ (Mouret & Doncieux,
2010) and uses on the DART simulator (Lee et al., 2018), while the auto-
encoders are coded and trained using the C++ API of PyTorch (Paszke et al.,
2019). For each QD task, all variants were run for 10 replications.

##  6 Results

Figure 2: Containers returned by the different variants (columns) for each
Quality-Diversity task (rows). Each dot corresponds to one policy present in
the container, after projection in the Behavioural Descriptor space
$\mathcal{B}_{\mathcal{T}}$ of the QD task. The colour of each dot is
representative of the performance score of the policy. If the container result
from a Hand-Coded variant that is not associated to the task (e.g. the
container generated by HC-Nav for the Half-roll task), then the solutions are
shown in grey. Also, note that the BD space of Duty Factors (second row) has
six dimensions, the containers are presented using the same type of grids as
in the work of Cully et al. (2015).  Figure 3: Coverage per minimum
performance (presented in section 5.4), considered with all downstream tasks
under study. If the container results from a Hand-Coded variant that is not
associated to the task (e.g. the container generated by HC-Forw for the
Navigation task), then the performance of solutions are not considered, and
only the total coverage is presented. The vertical dashed line shows where to
read the total coverage (without any minimal performance) achieved by each
variant. For each variant, the bold line represents the median coverage; and
the shaded area indicates the interquartile range.

Our objective is to show that AURORA discovers diversity in all directions of
the BD space. To that end, we take the containers returned by the different
algorithms, and we project them in the BD spaces associated to each Quality-
Diversity (QD) task. Then we evaluate the achieved coverage in the BD space by
considering the coverage for several minimum performance scores.

From Figure 3, it can be noticed that AURORA systematically obtains a lower
total coverage than the variant using the hand-coded BD of the task (e.g. HC-
Forw for the Forward task). Such variants use a BD that is well-adapted to the
QD problem to solve; but this kind of BD requires human expertise to be
defined.

AURORA can also be compared to Hand-coded containers after a projection in the
BD spaces of other tasks, such as the container from HC-Nav projected in the
BD space of the Moving Forward task. We notice that those kinds of containers
(shown in grey in Figs 2 and 3) achieve a coverage that is at most equivalent,
if not worse, compared to the containers of AURORA. For example, in the Half-
roll task, the Hand-coded baselines HC-Nav and HC-Forw do not find any
solution making the hexapod fall on its back; whereas AURORA discovers such
behaviours (see in corners of Half-roll containers in Fig. 2). In the end,
AURORA manages to find containers of behaviours exhibiting diversity in all BD
spaces.

The MeS variant, whose BD corresponds to the mean of each stream of sensory
data, does not provide the same level of efficiency as AURORA. In the three
tasks, AURORA obtains a coverage that is higher or equal to the coverage
obtained for MeS (Fig. 3). When considering high minimum performance, the gap
between the coverages from AURORA and MeS is even more apparent. This suggests
that the automatic definition of an unsupervised BD may improve the
performance upon hand-defined BDs, even if those are calculated considering
all full states of the robot. In the end, by considering the entire full-state
trajectory of the agent, AURORA can learn a large diversity of skills in an
unsupervised manner, which is more general than any of the compared methods.

##  7 Conclusion and Future Work

In this paper, we provided an additional analysis of AURORA, a QD algorithm
that automatically defines its BDs. In that analysis, we studied what kind of
behaviours emerge when AURORA encodes the full-state trajectory of an hexapod
robot. Our experimental evaluation demonstrated that AURORA autonomously finds
diverse types of behaviours such as: navigating to different positions, using
its legs in different manners, and performing half-rolls.

In order to limit the computational complexity of AURORA, the container is
forced to have a limited size (usually of the order of magnitude of 10,000
policies). This property may prevent the algorithm from exploring in detail
all the different types of behaviours in open-ended environments. For example,
it would be interesting to simultaneously obtain a complete behavioural sub-
container specialised in half-rolls, and another sub-container specialised in
navigation. This problem has been efficiently addressed by Etcheverry et al.
(2020) with IMGEP-HOLMES, an algorithm that manages to discover several
specialised niches of behaviours. It would be interesting to study how AURORA
could integrate the same mechanisms as IMGEP-HOLMES to handle open-endedness.
It would also be relevant to try our approach on more open-ended problems,
where the robot can interact with various objects and other robots.

#### Acknowledgments

This work was supported by the Engineering and Physical Sciences Research
Council (EPSRC) grant EP/V006673/1 project REcoVER. We thank the members of
the Adaptive and Intelligent Robotics Lab for their very valuable comments.

## References

  * Akkaya et al. (2019) Ilge Akkaya, Marcin Andrychowicz, Maciek Chociej, Mateusz Litwin, Bob McGrew, Arthur Petron, Alex Paino, Matthias Plappert, Glenn Powell, Raphael Ribas, et al.  Solving rubik’s cube with a robot hand.  _arXiv preprint arXiv:1910.07113_ , 2019.
  * Bentley (1975) Jon Louis Bentley.  Multidimensional binary search trees used for associative searching.  _Communications of the ACM_ , 18(9):509–517, 1975\.
  * Bossens et al. (2020) David M Bossens, Jean-Baptiste Mouret, and Danesh Tarapore.  Learning behaviour-performance maps with meta-evolution.  In _Proceedings of the 2020 Genetic and Evolutionary Computation Conference_ , pp. 49–57, 2020.
  * Chatzilygeroudis et al. (2018) Konstantinos Chatzilygeroudis, Vassilis Vassiliades, and Jean-Baptiste Mouret.  Reset-free trial-and-error learning for robot damage recovery.  _Robotics and Autonomous Systems_ , 100:236–250, 2018.
  * Cover & Thomas (2012) Thomas M Cover and Joy A Thomas.  _Elements of information theory_.  John Wiley & Sons, 2012.
  * Cully (2019) Antoine Cully.  Autonomous skill discovery with quality-diversity and unsupervised descriptors.  In _GECCO 2019 - Proceedings of the 2019 Genetic and Evolutionary Computation Conference_ , pp. 81–89. Association for Computing Machinery, Inc, jul 2019.  ISBN 9781450361118.
  * Cully & Demiris (2018) Antoine Cully and Yiannis Demiris.  Quality and Diversity Optimization: A Unifying Modular Framework.  _IEEE Transactions on Evolutionary Computation_ , 22(2):245–259, apr 2018.  ISSN 1089778X.
  * Cully & Mouret (2013) Antoine Cully and Jean Baptiste Mouret.  Behavioral repertoire learning in robotics.  In _GECCO 2013 - Proceedings of the 2013 Genetic and Evolutionary Computation Conference_ , pp. 175–182, New York, New York, USA, 2013. ACM Press.  ISBN 9781450319638.
  * Cully et al. (2015) Antoine Cully, Jeff Clune, Danesh Tarapore, and Jean-Baptiste Mouret.  Robots that can adapt like animals.  _Nature_ , 521(7553):503–507, 2015.
  * Deb & Agrawal (1999) Kalyanmoy Deb and Samir Agrawal.  A niched-penalty approach for constraint handling in genetic algorithms.  In _Artificial Neural Nets and Genetic Algorithms_ , pp. 235–243. Springer, 1999.
  * Duarte et al. (2016) Miguel Duarte, Jorge Gomes, Sancho Moura Oliveira, and Anders Lyhne Christensen.  Evorbc: evolutionary repertoire-based control for robots with arbitrary locomotion complexity.  In _Proceedings of the Genetic and Evolutionary Computation Conference 2016_ , pp. 93–100. ACM, 2016.
  * Etcheverry et al. (2020) Mayalen Etcheverry, Clément Moulin-Frier, and Pierre-Yves Oudeyer.  Hierarchically organized latent modules for exploratory search in morphogenetic systems.  _Advances in Neural Information Processing Systems_ , 33:4846–4859, 2020.
  * Eysenbach et al. (2018) Benjamin Eysenbach, Abhishek Gupta, Julian Ibarz, and Sergey Levine.  Diversity is all you need: Learning skills without a reward function.  _arXiv preprint arXiv:1802.06070_ , 2018.
  * Gregor et al. (2016) Karol Gregor, Danilo Jimenez Rezende, and Daan Wierstra.  Variational intrinsic control.  _arXiv preprint arXiv:1611.07507_ , 2016.
  * Grillotti & Cully (2021) Luca Grillotti and Antoine Cully.  Unsupervised behaviour discovery with quality-diversity optimisation.  _arXiv preprint arXiv:2106.05648_ , 2021.
  * Hwangbo et al. (2019) Jemin Hwangbo, Joonho Lee, Alexey Dosovitskiy, Dario Bellicoso, Vassilios Tsounis, Vladlen Koltun, and Marco Hutter.  Learning agile and dynamic motor skills for legged robots.  _Science Robotics_ , 4(26):eaau5872, 2019.
  * Johns et al. (2016) Edward Johns, Stefan Leutenegger, and Andrew J Davison.  Deep learning a grasp function for grasping under gripper pose uncertainty.  In _2016 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)_ , pp. 4461–4468. IEEE, 2016.
  * Kaushik et al. (2020) Rituraj Kaushik, Pierre Desreumaux, and Jean-Baptiste Mouret.  Adaptive prior selection for repertoire-based online adaptation in robotics.  _Frontiers in Robotics and AI_ , pp. 151, 2020.
  * Kingma & Ba (2014) Diederik P Kingma and Jimmy Ba.  Adam: A method for stochastic optimization.  _arXiv preprint arXiv:1412.6980_ , 2014.
  * Lee et al. (2018) Jeongseok Lee, Michael X Grey, Sehoon Ha, Tobias Kunz, Sumit Jain, Yuting Ye, Siddhartha S Srinivasa, Mike Stilman, and C Karen Liu.  Dart: Dynamic animation and robotics toolkit.  _Journal of Open Source Software_ , 3(22):500, 2018.
  * Liapis et al. (2013) Antonios Liapis, Héctor P Martínez, Julian Togelius, and Georgios N Yannakakis.  Transforming Exploratory Creativity with DeLeNoX.  Technical report, 2013.
  * Liu & Abbeel (2021a) Hao Liu and Pieter Abbeel.  Aps: Active pretraining with successor features.  In _International Conference on Machine Learning_ , pp. 6736–6747. PMLR, 2021a.
  * Liu & Abbeel (2021b) Hao Liu and Pieter Abbeel.  Behavior from the void: Unsupervised active pre-training.  _Advances in Neural Information Processing Systems_ , 34, 2021b.
  * Masci et al. (2011) Jonathan Masci, Ueli Meier, Dan Cireşan, and Jürgen Schmidhuber.  Stacked convolutional auto-encoders for hierarchical feature extraction.  In _International conference on artificial neural networks_ , pp. 52–59. Springer, 2011.
  * Meyerson et al. (2016) Elliot Meyerson, Joel Lehman, and Risto Miikkulainen.  Learning behavior characterizations for novelty search.  In _Proceedings of the Genetic and Evolutionary Computation Conference 2016_ , pp. 149–156, 2016.
  * Mouret & Doncieux (2010) J.-B. Mouret and S. Doncieux.  SFERESv2: Evolvin’ in the multi-core world.  In _Proc. of Congress on Evolutionary Computation (CEC)_ , pp. 4079–4086, 2010.
  * Mouret & Clune (2015) Jean-Baptiste Mouret and Jeff Clune.  Illuminating search spaces by mapping elites.  apr 2015.
  * Paolo et al. (2020) Giuseppe Paolo, Alban Laflaquiere, Alexandre Coninx, and Stephane Doncieux.  Unsupervised learning and exploration of reachable outcome space.  In _2020 IEEE International Conference on Robotics and Automation (ICRA)_ , pp. 2379–2385. IEEE, 2020.
  * Paszke et al. (2019) Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, et al.  Pytorch: An imperative style, high-performance deep learning library.  In _Advances in neural information processing systems_ , pp. 8026–8037, 2019.
  * Sharma et al. (2019) Archit Sharma, Shixiang Gu, Sergey Levine, Vikash Kumar, and Karol Hausman.  Dynamics-aware unsupervised discovery of skills.  _arXiv preprint arXiv:1907.01657_ , 2019.


# The class of Krasner hyperfields is not elementary

Piotr Błaszkiewicz♣ ♠Instytut Matematyczny
Uniwersytet Wrocławski
Wrocław
Poland<EMAIL_ADDRESS>and  Piotr Kowalski♠ ♠Instytut
Matematyczny
Uniwersytet Wrocławski
Wrocław
Poland<EMAIL_ADDRESS>http://www.math.uni.wroc.pl/~pkowa/

###### Abstract.

We show that the class of Krasner hyperfields is not elementary. To show this,
we determine the rational rank of quotients of multiplicative groups in field
extensions. Our argument uses Chebotarev’s density theorem. We also discuss
some related questions.

♣ Supported by the Narodowe Centrum Nauki grant no. 2021/43/B/ST1/00405.

♠ Supported by the Narodowe Centrum Nauki grant no. 2021/43/B/ST1/00405.

2020 Mathematics Subject Classification Primary 03C60; Secondary 16Y20.

Key words and phrases. Hyperfield, Elementary class, Chebotarev’s density
theorem.

##  1\. Introduction

The notion of a hyperfield (_hypercorps_) was introduced first by Marc Krasner
in [12] as a tool to study valued fields. In his later paper [13], he
introduced the quotient construction of a hyperfield from a given field and a
subgroup of its multiplicative group (see Theorem 2.2).

The question whether all hyperfields come from this quotient construction has
been an open problem until the example of Massouros, who showed in [17] that
it is not the case. Nevertheless, the class of Krasner hyperfields (i.e.,
hyperfields obtained by this quotient construction) contains a lot of known
examples of hyperfields. Among them, there are the hyperfields known as RV-
sorts, which are studied in the model theory of valued fields. They were first
introduced by Joseph Flenner in his PhD thesis [7] as a tool to obtain
(relative) quantifier elimination for valued fields. Flenner proved that RV-
sorts are bi-interpretable with $amc$-structures (three sorted structures)
introduced by Franz-Viktor Kuhlmann in [14]. Currently, hyperfields in the
form of RV-sorts are one of the main objects used to study model theory of
valued fields (see e.g. [2, 15, 21, 22]).

In view of the usefulness of Krasner hyperfields for the model theory of
valued fields discussed above, we were motivated to study model theoretical
properties of Krasner hyperfields themselves. Since the definition of Krasner
hyperfields is purely algebraical, the first question we faced was: “Is the
class of Krasner hyperfields elementary?”. Based on the results of Alain
Connes and Caterina Consani from [3], we show in this paper that this class is
_not_ elementary.

The paper is organized as follows. In Section 2, we collect the necessary
facts and results about hyperfields as well as results from field theory. In
Section 3, we prove our main algebraic result (Theorem 3.3) and use it to show
that the class of Krasner hyperfields is not elementary (Corollary 3.9). In
Section 4, we quickly state and prove this algebraic result in its proper
generality (Theorem 4.1) and discuss some model-theoretical problems related
with hyperfields and the algebraic methods used in this paper (Question 4.2
and Conjecture 4.3).

We would like to thank Franz-Viktor Kuhlmann for his careful reading of this
paper and his suggestions for improvement. Last but not least we would like to
thank the members of the Wrocław model theory seminar for their insightful
questions and comments concerning this paper.

##  2\. Preliminaries

In this section, we introduce the necessary notions we are going to use
throughout this paper (or we cite the necessary sources). Further we will
present the results from the paper of Alain Connes and Caterina Consani [3],
where (among other things) they studied connections between Krasner
hyperfields and projective geometries. Finally, we state the Chebotarev’s
density theorem which will play a crucial role in the arguments in Section 3.

Not everything from this section is directly needed for the arguments in
Section 3, e.g. Theorem 2.11, Facts 2.13, 2.14, or Remark 2.16 will not be
used directly. However, we hope that these extra results provide a greater
picture and they also show how to avoid possible “wrong paths” in the main
argument.

###  2.1. Hyperfields

The notion of a hyperfield, as one could expect, generalises the one of a
field. The twist is that the addition is a multivalued operation, so instead
of an element it returns a nonempty set.

######  Definition 2.1.

A hyperfield is a tuple $(\mathcal{H},+,\cdot,0,1)$ where
$(\mathcal{H},\cdot,1)$ is an abelian group and

$+:\mathcal{H}\times\mathcal{H}\rightarrow\mathcal{P}(\mathcal{H})\setminus\\{\emptyset\\}$

satisfies the following axioms, where $x,y,z\in\mathcal{H}$ and $+,\cdot$ are
naturally extended to subsets of $\mathcal{H}$:

  * •

$x+y=y+x$ (commutativity),

  * •

$(x+y)+z=x+(y+z)$ (associativity),

  * •

for each $x\in\mathcal{H}$, there is a unique $-x\in\mathcal{H}$ such that
$0\in x+(-x)$ (unique inverse),

  * •

$z\in x+y\Rightarrow y\in z+(-x)$ (reversibility),

  * •

$x+0=\\{x\\}$ (neutral element),

  * •

$z\cdot(x+y)=z\cdot x+z\cdot y$ (distributivity).

Note that every field can be viewed as a hyperfield in the obvious way. For
more details and preliminary notions concerning hyperfields (such as
homomorphisms, hyperideals, etc.) we direct the reader to [4, 10, 16].

We state now the theorem of Krasner which was mentioned in the introduction.

######  Theorem 2.2.

Let $K$ be a field and $G$ a subgroup of $K^{\times}$. The quotient
$K^{\times}/G$ together with an extra element $0$ and $+,\cdot$ defined as:

  * •

$aG\cdot bG:=abG$,

  * •

$aG+bG:=\\{(x+y)G\mid x\in aG,y\in bG\\}$

forms a hyperfield, where $1=G$.

######  Notation 2.3.

We will abbreviate $(K^{\times}/G)\cup\\{0\\}$ from Theorem 2.2 as $K/G$.

######  Definition 2.4.

If a hyperfield $\mathcal{H}$ is isomorphic to $K/G$ (as in Notation 2.3),
then we call it a Krasner hyperfield.

###  2.2. Projective geometries and CC-hyperfields

The material of this subsection comes from [3]. We introduce first the
definition of $CC$-hyperfields.

######  Definition 2.5.

We call a hyperfield $(\mathcal{H},+,\cdot,0,1)$ a $CC$-hyperfield, if

$\forall x\in\mathcal{H}\ \ x+x=\\{x,0\\}.$

######  Remark 2.6.

The name “$CC$-hyperfield” does not appear anywhere except this paper. In
particular, it was not used in [3] where hyperfields with this property were
studied. However, the name used in [3] is quite technical (“hyperfield
extensions of the Krasner hyperfield $\mathbf{K}$”) and, for simplicity, we
decided to refer to this class of hyperfields as $CC$-hyperfields.

There is the following nice description of Krasner CC-hyperfields.

######  Proposition 2.7 (Proposition 2.7 in [3]).

Let $K$ be a field and $G$ be a subgroup of $K^{\times}$. Assume that
$G\neq\\{1\\}$. Then the hyperfield $K/G$ is a $CC$-hyperfield if and only if
$\\{0\\}\cup G$ is a subfield of $K$.

The class of CC-hyperfields is also closely related to _projective geometries_
(in the sense of _incidence geometry_ , for the necessary notions see e.g.
[1]).

######  Proposition 2.8 ([3] Proposition 3.1).

If $\mathcal{H}$ is a CC-hyperfield, then there is a unique projective
geometry on $\mathcal{H}\setminus\\{0\\}$ such that for distinct
$x,y\in\mathcal{H}\setminus\\{0\\}$, the unique line through $x$ and $y$
coincides with $\\{x,y\\}\cup x+y$.

######  Notation 2.9.

We will denote the above projective geometry by $\mathcal{P}_{\mathcal{H}}$.

######  Remark 2.10.

If $\mathcal{H}$ is a Krasner CC-hyperfield, then the corresponding projective
geometry $\mathcal{P}_{\mathcal{H}}$ is the classical one, which we will see
below. By Proposition 2.8, $\mathcal{H}=L/K^{\times}$, where $K$ is a subfield
of $L$. Then we can view $L$ as a vector space over $K$ and consider the
classical projective geometry associated to this vector space. This geometry
happens to be exactly the projective geometry associated to the CC-hyperfield
$L/K^{\times}$. In particular, such a projective geometry is always
_Desarguesian_ and we have

$\dim\left(\mathcal{P}_{\mathcal{H}}\right)+1=[L:K].$

We finish this subsection with a result from [3] which will tell us later that
we need to focus on Krasner CC-hyperfields of dimension one, where by the
dimension of a CC-hyperfield, we always mean the dimension of its associated
projective geometry.

######  Theorem 2.11 ([3] Theorem 3.8).

Let $\mathcal{H}$ be a $CC$ hyperfield. Assume that the projective geometry
$\mathcal{P}_{\mathcal{H}}$ is Desarguesian and of dimension at least 2. Then
there exists a unique pair $(L,K)$ where $L$ is a field, and $K$ is its
subfield such that

$\mathcal{H}=L/K^{\times}.$

###  2.3. Model theory

In this subsection, we specify the model-theoretical set-up which is needed to
work with hyperfields. We also show several reduction results.

We start with specifying the first-order language of hyperfields.

######  Definition 2.12.

Let us set the language of hyperfields as the tuple
$(\oplus,\ominus,\odot,^{-1},\underline{0},\underline{1})$, where:

  * •

$\odot$ is a binary function symbol interpreted as a multiplication,

  * •

-1 is unary function symbol interpreted as a multiplicative inverse,

  * •

$\oplus$ is a ternary relation symbol encoding the hyperaddition (so, in a
hyperfield we will have: $\oplus(x,y,z)$ if and only if $z\in x+y$),

  * •

$\ominus$ is a unary function symbol encoding the additive inverse (so,
$\oplus(x,\ominus x,0)$ holds in a hyperfield),

  * •

$\underline{1}$ and $\underline{0}$ are constant symbols corresponding to the
neutral elements of the multiplication and the hyperaddition respectively.

Clearly, the class of hyperfields can be first-order axiomatized in the
language above. Let us state the following well-known result.

######  Fact 2.13.

The class $\mathcal{C}$ of structures (in a fixed language) is elementary if
and only if $\mathcal{C}$ is closed under elementary equivalence and under
ultraproducts.

As a simple consequence of Łoś’s theorem, one obtains the following.

######  Fact 2.14.

The class of Krasner factor hyperfields is closed under ultraproducts.

Therefore, we will aim to show that the class of Krasner hyperfields is not
closed under elementary equivalence. We see below that we can restrict
ourselves to the class of Krasner CC-hyperfields.

######  Lemma 2.15.

If the class of Krasner hyperfields is elementary, then the class of Krasner
CC-hyperfields is elementary.

###### Proof.

It is obvious, since the condition $(\forall x)(x+x=\\{0\\})$ is clearly
definable in the language from Definition 2.12. ∎

######  Remark 2.16.

All the assumptions from Theorem 2.11 can be expressed as first-order
sentences in the language of hyperfields introduced above (using the explicit
definition of the associated projective geometry from Proposition 2.8). Hence,
we obtain that the class of Krasner $CC$-hyperfields of dimension at least 2
_is_ elementary.

Because of Remark 2.16, we need to focus on one-dimensional Krasner CC-
hyperfields. For convenience, we give names to the following two classes.

######  Notation 2.17.

  1. (1)

Let $\mathcal{K}$ denote the class of Krasner CC-hyperfields of dimension one.

  2. (2)

Let $\mathcal{K}^{\times}$ denote the class of groups which are of the form
$L^{\times}/K^{\times}$, where $K\subseteq L$ is a field extension of degree
2.

The next observation explains why the class of groups from Notation 2.17(2) is
important for us.

######  Fact 2.18.

Let $\mathcal{H}\in\mathcal{K}$. Then we have the following.

  1. (1)

$\mathcal{H}$ is isomorphic to $L/K^{\times}$ where $L$ is a field, $K$ is its
subfield and $[L:K]=2$.

  2. (2)

The hyperaddition in $\mathcal{H}$ is given by the following formula:

$\displaystyle x+y=\mathcal{H}\setminus\\{0\\}\mbox{ for x $\neq$ y $\neq$
0,}$ $\displaystyle x+x=\\{x,0\\},$ $\displaystyle x+0=0+x=\\{x\\},$

so it is definable in the language $\\{\underline{0}\\}$ (just one constant
symbol).

###### Proof.

Item (1) follows from Remark 2.10, since $\dim(\mathcal{P}_{\mathcal{H}})=1$
if and only if $\mathcal{H}$ comes from a field extension of degree 2.

Item (2) follows again from Remark 2.10 (and Proposition 2.8), since
$\dim(\mathcal{P}_{\mathcal{H}})=1$ implies that there is only one line in the
projective geometry $\mathcal{P}_{\mathcal{H}}$ and this line is the whole
space. ∎

We need one more result from [3] saying that some hyperfields are definable
just from their multiplicative structure. Such a phenomenon is impossible for
fields and this is the base of our proof of the main result of this paper
(Corollary 3.9).

######  Fact 2.19 (Proposition 3.6 in [3]).

Let $(G,\cdot)$ be a commutative group (written multiplicatively) of order at
least 4. We define $\mathcal{H}_{G}$ as $G\cup\\{0\\}$, where $0$ is a new
symbol and extend the group operation on $G$ to the commutative monoid
structure on $\mathcal{H}_{G}$ by setting $x\cdot 0=0=0\cdot x$. If we define
the hyperaddition $+$ on $\mathcal{H}_{G}$ as in Fact 2.18(2), then
$(\mathcal{H}_{G},+,\cdot)\in\mathcal{K}$ (see Notation 2.17(1)).

As a consequence, we directly obtain the following.

######  Lemma 2.20.

If the class of Krasner CC-hyperfields is elementary, then the class
$\mathcal{K}$ is elementary.

###### Proof.

By Facts 2.18 and 2.19, if $\mathcal{H}$ is a Krasner CC-hyperfields, then
$\mathcal{H}\in\mathcal{K}$ if and only if the hyperaddition in $\mathcal{K}$
is given by the formula as in Fact 2.18(2), which is clearly a first-order
condition. ∎

The following easy results outline the further connections between the classes
$\mathcal{K}$ and $\mathcal{K}^{\times}$.

######  Lemma 2.21.

Let $G$ and $H$ be commutative groups and $\mathcal{H}_{G},\mathcal{H}_{H}$ be
the corresponding hyperfields as in Fact 2.19. If $G$ and $H$ are elementarily
equivalent (as groups), then $\mathcal{H}_{G}$ and $\mathcal{H}_{H}$ are
elementarily equivalent (as hyperfields).

###### Proof.

Assume that $(G,\cdot)\equiv(H,\cdot)$. By the uniform definition of the
monoid operation in $\mathcal{H}_{G},\mathcal{H}_{H}$, we get that
$(\mathcal{H}_{G},\cdot)\equiv(\mathcal{H}_{H},\cdot)$. Since the
hyperaddition in $\mathcal{H}_{G},\mathcal{H}_{H}$ is defined by the same
formula in the monoid language (we actually only need the extra constant as in
Fact 2.18), we get that
$(\mathcal{H}_{G},+,\cdot)\equiv(\mathcal{H}_{H},+,\cdot)$. ∎

######  Lemma 2.22.

If the class $\mathcal{K}$ is elementary, then the class
$\mathcal{K^{\times}}$ is closed under elementary equivalence.

###### Proof.

Assume that the class $\mathcal{K}$ is elementary. Let us take
$G\in\mathcal{K^{\times}}$, so there is a hyperfield
$\mathcal{H}\in\mathcal{K}$ such that $G$ is the multiplicative group of
$\mathcal{H}$. By Facts 2.18 and 2.19, we get that
$\mathcal{H}=\mathcal{H}_{G}$ (as hyperfields). We also take a group $H$ such
that $G\equiv H$. By Lemma 2.21, we get that
$\mathcal{H}_{H}\equiv\mathcal{H}_{G}=\mathcal{H}$. Since the class
$\mathcal{K}$ is elementary, we obtain that $\mathcal{H}_{H}\in\mathcal{K}$.
Therefore, $H\in\mathcal{K}^{\times}$, which we needed to show. ∎

With those results at hand, it is enough to show that the class
$\mathcal{K}^{\times}$ is not closed under elementary equivalence, which we
will do in Section 3. In the next subsection, we will describe the necessary
algebraic tools to obtain this result.

###  2.4. Chebotarev’s density theorem

We will use one classical result which is a consequence of Chebotarev’s
density theorem. We need some terminology first (see [8, Chapter 6]).

Assume that $K\subseteq L$ is a finite Galois extension, where $K$ is either a
number field or the field of rational functions ${\mathds{F}}_{q}(X)$ over a
finite field. Let $\mathcal{O}_{K},\mathcal{O}_{L}$ be the integral closures
of ${\mathds{Z}}$ or ${\mathds{F}}_{q}[X]$ in $K$ and $L$ respectively. Then
$\mathcal{O}_{K}$ and $\mathcal{O}_{L}$ are Dedekind rings, hence each non-
zero ideal decomposes uniquely into a finite product of maximal ideals. It is
also well-known that if $P$ is a maximal ideal of $\mathcal{O}_{K}$, then we
have $P\mathcal{O}_{L}=Q_{1}\cdot\ldots\cdot Q_{k}$, where $Q_{i}$ are maximal
ideals of $\mathcal{O}_{L}$ and $k\leqslant[L:K]$. If $k>1$ and the ideals
$Q_{1},\ldots,Q_{k}$ are pairwise distinct, then we say that $P$ _splits_ in
$L$. If $P$ splits in $L$, and if $k=[L:K]$, then we say that $P$ _splits
completely_ in $L$. In the case of a degree two extension, which is the
situation corresponding to the class $\mathcal{K}$ from Notation 2.17(1), the
notions of splitting and splitting completely coincide.

We need the following consequence of Chebotarev’s density theorem.

######  Theorem 2.23.

Let $K\subseteq L$ be a finite Galois extension as above. Then there are
infinitely many maximal ideals in $\mathcal{O}_{K}$ which split completely in
$L$.

###### Proof.

This directly follows from [8, Exercise 5(a), page 129] together with [8,
Theorem 6.3.1] (Chebotarev’s density theorem) and its interpretation stated in
the paragraph before it (the sets of density $0$ are finite). ∎

##  3\. Rational rank

In this section, we prove the main results of this paper (Theorem 3.3 and
Corollary 3.9).

We need the following notion (see [6, Section 3.4]).

######  Definition 3.1.

The _rational rank_ of a commutative group $A$ is the cardinality of a maximal
${\mathds{Z}}$-linearly independent subset of $A$. Following [6], we denote it
by $\operatorname{rr}(A)$.

######  Remark 3.2.

Let $A$ be a commutative group.

  1. (1)

It is easy to see that we have (see [6, Section 3.4]):

$\operatorname{rr}(A)=\dim_{{\mathds{Q}}}\left(A\otimes_{{\mathds{Z}}}{\mathds{Q}}\right).$

  2. (2)

If $A_{0}\leqslant A$, then we have (see [6, Section 3.4]):

$\operatorname{rr}(A)=\operatorname{rr}(A_{0})+\operatorname{rr}\left(A/A_{0}\right).$

  3. (3)

Other names as “rank” or “Prüfer rank” or “torsion-free rank” are sometimes
used in this context as well.

Our main algebraic result is the following (see Notation 2.17).

######  Theorem 3.3.

The rational rank of any $A\in\mathcal{K}^{\times}$ is either $0$ or infinite.

######  Remark 3.4.

This result will be generalized in Theorem 4.1 to a much wider class of
commutative groups coming from multiplicative groups of fields.

We proceed to the proof of Theorem 3.3. We start with a warm-up example which
will be also used in the main proof.

######  Example 3.5.

We will show here two particular cases of Theorem 3.3. For the proof of the
first one, we will use a well-known algebraic fact that is also a direct
consequence of Chebotarev’s theorem. In the proof of the second one,
Chebotarev’s theorem will not be needed.

  1. (1)

We have

$\operatorname{rr}({\mathds{Q}}[i]^{\times}/{\mathds{Q}}^{\times})=\aleph_{0}.$

###### Proof.

Let us recall that a prime number $p\in{\mathds{Z}}$ splits (equivalently in
this case: splits completely) in ${\mathds{Z}}[i]$ if and only if $p\equiv
1(\mbox{mod }4)$ and that there are infinitely many such primes (this is a
very special case of Theorem 2.23). Let us take an infinite sequence
$p_{1},p_{2},\ldots$ of prime numbers which split in ${\mathds{Z}}[i]$.

We have $p_{i}=r_{i}\overline{r_{i}}$, where $r_{i}$ is a prime element of
${\mathds{Z}}[i]$ and $\overline{r_{i}}$ is the complex conjugate of $r_{i}$.
Then $r_{1},\overline{r_{1}},r_{2},\overline{r_{2}},\ldots$ is a sequence of
pairwise non-associated prime elements of ${\mathds{Z}}[i]$. We will show that
the cosets $r_{1}{\mathds{Q}}^{\times},r_{2}{\mathds{Q}}^{\times},\ldots$ are
${\mathds{Z}}$-independent in
${\mathds{Q}}[i]^{\times}/{\mathds{Q}}^{\times}$.

Assume not, so there is a non-zero tuple
$(n_{1},n_{2},...,n_{k})\in{\mathds{Z}}^{k}$ such that
$r_{1}^{n_{1}}r_{2}^{n_{2}}\ldots r_{k}^{n_{k}}\in{\mathds{Q}}^{\times}$
(witnessing that
$r_{1}{\mathds{Q}}^{\times},r_{2}{\mathds{Q}}^{\times},..,r_{k}{\mathds{Q}}^{\times}$
are not ${\mathds{Z}}$-independent in
${\mathds{Q}}[i]^{\times}/{\mathds{Q}}^{\times}$). We have:

$r_{1}^{n_{1}}r_{2}^{n_{2}}...r_{k}^{n_{k}}=\overline{r_{1}}^{n_{1}}\overline{r_{2}}^{n_{2}}...\overline{r_{k}}^{n_{k}},$

which contradicts the unique factorization in ${\mathds{Z}}[i]$ (after
rearranging the displayed equation in such a way that all the exponents are
positive). ∎

  2. (2)

We have

$\operatorname{rr}({\mathds{Q}}(X)^{\times}/{\mathds{Q}}(X^{2})^{\times})=\aleph_{0}.$

###### Proof.

The proof here is analogous to the proof in item $(1)$. We take pairwise
distinct rational numbers $a_{1},a_{2},\ldots$ and substitute:

${\mathds{Z}}\rightsquigarrow{\mathds{Q}}[X^{2}],\ \
{\mathds{Z}}[i]\rightsquigarrow{\mathds{Q}}[X],\ \ p_{i}\rightsquigarrow
X^{2}-a_{i}^{2},\ \ r_{i}\rightsquigarrow X-a_{i},$

where the map ${\mathds{Q}}(X)\ni f\mapsto f(-X)\in{\mathds{Q}}(X)$ plays the
role of the complex conjugation. ∎

We will prove now several technical results which will be used in the proof of
Theorem 3.3.

######  Lemma 3.6.

Let $F\subseteq K$ be a field extension of degree $2$. Suppose that $F$ is
algebraic over a finite field or the extension $F\subseteq K$ is not Galois.
Then we have:

$\operatorname{rr}\left(K^{\times}/F^{\times}\right)=0.$

###### Proof.

If $F$ is algebraic over a finite field, then $K^{\times}$ is torsion.
Therefore, $K^{\times}/F^{\times}$ is torsion as well and
$\operatorname{rr}\left(K^{\times}/F^{\times}\right)=0.$

Assume that the extension $F\subseteq K$ is not Galois. Then
$\operatorname{char}(F)=2$ and the extension $F\subseteq K$ is purely
inseparable (since $[K:F]=2$). Therefore, $K^{2}\subseteq F$ (using again that
$[K:F]=2$), hence the group $K^{\times}/F^{\times}$ is torsion again. ∎

For the proof of Theorem 3.3, it is enough now to show the following:

######  Proposition 3.7.

Let $F\subseteq K$ be a Galois field extension of degree $2$ such that $F$ is
not algebraic over a finite field. Then the rational rank of
$K^{\times}/F^{\times}$ is infinite.

The next result allows as to reduce the proof of Proposition 3.7 to several
special cases.

######  Lemma 3.8.

Let $K\subseteq L$ be a field extension satisfying the assumption of
Proposition 3.7. Then either

  1. (1)

${\mathds{Q}}(X)^{\times}/{\mathds{Q}}(X^{2})^{\times}$, or

  2. (2)

$F^{\times}/E^{\times}$, where $E\subseteq F$ is a number field extension of
degree 2, or

  3. (3)

${\mathds{F}}_{p}(X)^{\times}/{\mathds{F}}_{p}(X^{2})^{\times}$ for $p\neq 2$,
or

  4. (4)

${\mathds{F}}_{2}(X)^{\times}/{\mathds{F}}_{2}(X^{2}+X)^{\times}$,

embeds into $L^{\times}/K^{\times}$.

###### Proof.

We consider several cases.

Case I $\operatorname{char}(K)=0$.

By a standard argument (which works whenever $\operatorname{char}(K)\neq 2$),
there is $\alpha\in L$ such that $\alpha^{2}\in K$ and $L=K(\alpha)$, so
$K\subseteq L$ is a Kummer extension. Since we are in the characteristic 0
case, we can assume that ${\mathds{Q}}$ is a subfield of $K$. By our choice of
$\alpha$, we have that ${\mathds{Q}}(\alpha)\cap K={\mathds{Q}}(\alpha^{2})$.
Therefore, ${\mathds{Q}}(\alpha)^{\times}/{\mathds{Q}}(\alpha^{2})^{\times}$
embeds into $L^{\times}/K^{\times}$. If $\alpha$ is transcendental over
${\mathds{Q}}$, then we are in the item $(1)$ situation. Otherwise, we are in
the item $(2)$ situation.

Case II $\operatorname{char}(K)=p>2$.

As in Case I, there is $\alpha\in L$ such that $\alpha^{2}\in K$ and
$L=K(\alpha)$, , so $K\subseteq L$ is a Kummer extension again. We can assume
that ${\mathds{F}}_{p}$ is a subfield of $K$. By our assumptions, there is
$t\in K$ which is transcendental over ${\mathds{F}}_{p}$. If $\alpha$ is
algebraic over ${\mathds{F}}_{p}$, then we replace $\alpha$ with $t\alpha$. If
$\alpha$ is transcendental over ${\mathds{F}}_{p}$, then we keep $\alpha$ as
it is. After this possible replacement, we moreover obtain that $\alpha$ is
transcendental over ${\mathds{F}}_{p}$. As in Case I,
${\mathds{F}}_{p}(\alpha)^{\times}/{\mathds{F}}_{p}(\alpha^{2})^{\times}$
embeds into $L^{\times}/K^{\times}$ and we are in the item $(3)$ situation.

Case III $\operatorname{char}(K)=2$.

Since the extension $K\subseteq L$ is Galois, by Artin-Schreier theory there
is $\alpha\in L$ such that $\alpha^{2}+\alpha\in K$ and $L=K(\alpha)$. We
proceed now as in Case II, however we (possibly) need to replace $\alpha$ with
$t+\alpha$ (rather than $t\alpha$). In the end, we are in the item $(4)$
situation. ∎

We are ready to show the main algebraic result of this section.

###### Proof of Prop. 3.7.

It is clear that if a commutative group $A$ embeds into a commutative group
$B$, then we have $\operatorname{rr}(A)\leqslant\operatorname{rr}(B)$.
Therefore, it is enough to consider the four cases given by the statement of
Lemma 3.8.

Case (1)
$\operatorname{rr}({\mathds{Q}}(X)^{\times}/{\mathds{Q}}(X^{2})^{\times})=\aleph_{0}$.

This is covered by Example 3.5(2).

Case (2)
$\operatorname{rr}(L^{\times}/K^{\times})=\aleph_{0}$, where $K\subseteq L$ is
a number field extension of degree 2.

This is a generalization of Example 3.5(1). Let $\mathcal{O}_{L}$ and
$\mathcal{O}_{K}$ denote the rings of algebraic integers of $L$ and $K$
respectively (as in Section 2.4) and let
$\operatorname{Gal}(L/K)=\\{\operatorname{id},\sigma\\}$.

By Theorem 2.23, there is an infinite sequence $P_{1},P_{2},\ldots$ of prime
ideals in $\mathcal{O}_{K}$ which split completely in $L$ (as we remarked
earlier, in the case of the degree 2 extension, splitting implies splitting
completely). Then for each $i$, we have:

$P_{i}=Q_{i}\sigma(Q_{i}),$

where $Q_{i}$ is a maximal ideal of $\mathcal{O}_{L}$ and all the ideals
$Q_{1},\sigma(Q_{1}),Q_{2},\sigma(Q_{2}),\ldots$ are pairwise distinct.

Using the Prime Avoidance Lemma (see e.g. [5, Section 3.2]), we choose
$a_{1},a_{2},\ldots\in\mathcal{O}_{L}$ such that

  * •

$a_{1}\in Q_{1}\setminus\sigma(Q_{1})$,

  * •

$a_{2}\in Q_{2}\setminus(Q_{1}\cup\sigma(Q_{1})\cup\sigma(Q_{2}))$,
and in general:

  * •

$a_{i}\in Q_{i}\setminus(Q_{1}\cup\ldots\cup
Q_{i-1}\cup\sigma(Q_{1})\cup\ldots\cup\sigma(Q_{i}))$.

Similarly as in Example 3.5(1) (although without using the unique
decomposition), we will show that the cosets
$a_{1}K^{\times},a_{2}K^{\times},\ldots$ are ${\mathds{Z}}$-independent in
$L^{\times}/K^{\times}$.

Assume not, so there is a nonzero tuple
$(n_{1},\ldots,n_{k})\in{\mathds{Z}}^{k}$ such that $a_{1}^{n_{1}}\ldots
a_{k}^{n_{k}}\in K^{\times}$ (witnessing that
$a_{1}K^{\times},\ldots,a_{k}K^{\times}$ are not ${\mathds{Z}}$-independent in
$L^{\times}/K^{\times}$). Since the tuple $(n_{1},\ldots,n_{k})$ is nonzero,
there is a smallest $r\leqslant k$, such that $n_{r}\neq 0$. Without loss of
generality, we can assume that $n_{r}>0$.

Since $a_{r}^{n_{r}}\ldots a_{k}^{n_{k}}\in K^{\times}$, we get

($*$) $a_{r}^{n_{r}}\ldots a_{k}^{n_{k}}=\sigma(a_{r}^{n_{r}}\ldots
a_{k}^{n_{k}})=\sigma(a_{r})^{n_{r}}\ldots\sigma(a_{k})^{n_{k}}.$

By our choice of the sequence $a_{1},a_{2},\ldots$, we have:

($**$) $a_{r}\in Q_{r},\ \ a_{r+1},\ldots,a_{{}_{k}}\notin Q_{r},\ \
\sigma(a_{r}),\ldots,\sigma(a_{k})\notin Q_{r}.$

Using $(*),(**)$ and doing a similar rearrangement as in Example 3.5(1), we
get that there are elements
$b\in\mathcal{O}_{L},c_{1},\ldots,c_{m}\in\mathcal{O}_{L}\setminus Q_{r}$ such
that

($***$) $a_{r}^{n_{r}}b=c_{1}\ldots c_{m}.$

Since $n_{r}>0$, $a_{r}\in Q_{r}$ and $Q_{r}$ is prime, the equality $(***)$
yields a contradiction, since its left-hand side belongs to $Q_{r}$ and its
right-hand side does not.

Case (3)
$\operatorname{rr}({\mathds{F}}_{p}(X)^{\times}/{\mathds{F}}_{p}(X^{2})^{\times})=\aleph_{0}$,
where $p\neq 2$.

The proof is almost the same as in Example 3.5(2), however now we substitute:

${\mathds{Z}}\rightsquigarrow{\mathds{F}}_{p}[X^{2}],\ \
{\mathds{Z}}[i]\rightsquigarrow{\mathds{F}}_{p}[X].$

The choice of irreducible polynomials in ${\mathds{F}}_{p}[X^{2}]$ which split
in ${\mathds{F}}_{p}[X]$ is not as straightforward as in Example 3.5(2). To
obtain such a sequence, we again use Theorem 2.23, which also covers Case (3).
With such a sequence at hand, we continue the proof as in Example 3.5(2).

Case (4)
$\operatorname{rr}({\mathds{F}}_{2}(X)^{\times}/{\mathds{F}}_{2}(X^{2}+X)^{\times})=\aleph_{0}$.

It is almost identical to the Case (3) situation, we just use the ring
${\mathds{F}}_{2}[X^{2}+X]$ instead of the ring ${\mathds{F}}_{p}[X^{2}]$. ∎

This concludes the proof of Proposition 3.7, and hence also the proof of
Theorem 3.3.

Our main model-theoretic result is below.

######  Corollary 3.9.

The class of Krasner hyperfields is not elementary.

###### Proof.

If the class of Krasner hyperfields is elementary, then the class
$\mathcal{K^{\times}}$ of groups (see Notation 2.17(2)) is closed under
elementary equivalence by Lemmas 2.15, 2.20, 2.22. We will show that this is
not the case.

Since ${\mathds{C}}^{\times}$ is divisible, we have:

$\mathcal{K}^{\times}\ni{\mathds{C}}^{\times}/{\mathds{R}}^{\times}\cong
A\oplus\bigoplus_{p}C_{p^{\infty}}\equiv{\mathds{Q}}\oplus\bigoplus_{p}C_{p^{\infty}},$

where $C_{p^{\infty}}$ is the Prüfer $p$-group and $A$ is a vector space over
${\mathds{Q}}$ of dimension continuum. The isomorphism above follows from the
classification of divisible commutative groups (see [11, Theorem 5 in Section
4]) and the elementary equivalence follows from the Szmielew’s description
(see [19]) of the theories of commutative groups (this elementary equivalence
can be also directly shown by taking ultraproducts).

However, the rational rank of ${\mathds{Q}}\oplus\bigoplus_{p}C_{p^{\infty}}$
is 1, so this group does not belong to $\mathcal{K}^{\times}$ by Theorem 3.3.
∎

##  4\. Generalization, conjecture and question

In this section we improve on Theorem 3.3 and we also discuss some model-
theoretical problems related with hyperfields and the algebraic methods used
in this paper.

We show now the following improvement of Theorem 3.3 to its proper generality.

######  Theorem 4.1.

Let $F\subseteq K$ be an arbitrary field extension. Then both
$\operatorname{rr}(F^{\times})$ and $\operatorname{rr}(K^{\times}/F^{\times})$
are $0$ or infinite.

###### Sketch of Proof.

If $F$ is algebraic over a finite field, then $F^{\times}$ is torsion, so
$\operatorname{rr}(F^{\times})=0$. If $F$ is not algebraic over a finite
field, then either ${\mathds{Q}}$ embeds into $F$ or ${\mathds{F}}_{p}(X)$
embeds into $F$ for some prime number $p$. So, for the case of
$\operatorname{rr}(F^{\times})$, it is enough to notice that

$\operatorname{rr}({\mathds{Q}}^{\times})=\operatorname{rr}({\mathds{F}}_{p}(X)^{\times})=\aleph_{0},$

which follows (as in Example 3.5) from the fact that both of these fields are
fraction fields of unique factorization domains with infinitely many pairwise
non-associated prime elements.

We move now to the case of a field extension $F\subseteq K$. If the extension
$F\subseteq K$ is not algebraic, we take a transcendental $t\in K$ and then
$\operatorname{rr}(F(t)^{\times}/F^{\times})$ is infinite by a similar
argument as above.

If the extension $F\subseteq K$ is purely inseparable or $K$ is contained in
the algebraic closure of a finite field, then
$\operatorname{rr}(K^{\times}/F^{\times})=0$ as in the proof of Lemma 3.6.

Therefore, we can assume that $F\subseteq K$ is a finite extension which is
not contained in the algebraic closure of a finite field and which is also not
purely inseparable. We aim to show that
$\operatorname{rr}(K^{\times}/F^{\times})$ is infinite. Let us take the field
tower $F\subseteq K_{0}\subseteq K$, where the first extension is separable
and non-trivial and the second one is purely inseparable. We have the
following exact sequence:

$1\to K_{0}^{\times}/F^{\times}\to K^{\times}/F^{\times}\to
K^{\times}/K_{0}^{\times}\to 1$

and we know that $\operatorname{rr}(K^{\times}/K_{0}^{\times})=0$, so using
Remark 3.2(2), we obtain that
$\operatorname{rr}(K_{0}^{\times}/F^{\times})=\operatorname{rr}(K^{\times}/F^{\times})$,
so we can moreover assume that $F\subseteq K$ is finite, separable, and
$F={\mathds{F}}_{p}(X)$ or $F$ is a number field. Let $F\subseteq L$ be the
normal closure of $F\subseteq K$ and $n:=[K:F]>1$. By Theorem 2.23, there are
infinitely many prime ideals $P_{1},P_{2},\ldots$ of $\mathcal{O}_{F}$ which
split completely in $\mathcal{O}_{L}$, so for each $i$, we also have
$P_{i}\mathcal{O}_{K}=Q_{i1}\ldots Q_{in}$, where $Q_{ij}$’s are maximal
ideals in $\mathcal{O}_{K}$. We take $a_{1},a_{2},\ldots\in\mathcal{O}_{K}$
such that for each $i$, we have:

$a_{i}\in
Q_{i1}\setminus\left(\bigcup_{j=1}^{i-1}Q_{j1}\cup\bigcup_{j=1}^{i}\bigcup_{k=2}^{l}Q_{jk}\right).$

Then the proof of the Case $(2)$ situation from Lemma 3.8 works after taking
$\sigma\in\operatorname{Gal}(L/F)$ such that $\sigma(a_{r})\notin
Q_{r1}\mathcal{O}_{L}$. ∎

Therefore (as in the proof of Corollary 3.9), the following class of groups:

$\\{K^{\times}\ |\ \text{$K$ is a field}\\}$

is _not_ elementary. Interestingly, a similar phenomenon appeared in [9] where
the authors consider model completeness of groups of rational points of
algebraic groups. One can ask the following.

######  Question 4.2.

Let $\mathbb{G}$ be a group scheme over ${\mathds{Z}}$. Are the following two
conditions on $\mathbb{G}$ equivalent?

  1. (1)

The class

$\\{\mathbb{G}(K)\ |\ \text{$K$ is a field}\\}$

is elementary.

  2. (2)

If $K$ is a model complete field, then $\mathbb{G}(K)$ is a model complete
group.

The multiplicative group scheme $\mathbb{G}_{\rm{m}}$ fails both items $(1)$
and $(2)$ above. On the other hand, semisimple or unipotent algebraic groups
seem to satisfy both these items, which is work in progress related to [9].
Therefore, we do not have counterexamples to the equivalence in Question 4.2.
Actually, if item $(1)$ holds, then (as in [9]) it is usually an important
step for proving that item $(2)$ holds. The fact that item $(1)$ holds for
certain simple algebraic groups follows from [18] and [20].

While trying to understand hyperfields (or any other structures) model-
theoretically, it is natural to ask first what are the “model-theoretically
simplest”, that is _strongly minimal_ , hyperfields. We propose the following.

######  Conjecture 4.3.

A hyperfield is strongly minimal if and only if it is either a strongly
minimal field (i.e., an algebraically closed field) or a hyperfield where the
hyperaddition is definable in the structure of its multiplicative group which
is strongly minimal.

Since any infinite commutative group can be expanded to a hyperfield where the
hyperaddition is definable just from one constant symbol (see Fact 2.19) there
are plenty hyperfields as after “or” in the conjecture above.

## References

  * [1] F. Buekenhout.  Handbook of Incidence Geometry: Buildings and Foundations.  Elsevier, 1995.
  * [2] A. Chernikov and P. Simon.  Henselian valued fields and inp-minimality.  The Journal of Symbolic Logic, 84(4):1510–1526, 2019.
  * [3] Alain Connes and Caterina Consani.  The hyperring of adèle classes.  Journal of Number Theory, 131(2):159–194, 2011.
  * [4] B. Davvaz and A. Salasi.  A realization of hyperrings.  Communications in Algebra, 34:4389–4400, 2006.
  * [5] D. Eisenbud.  Commutative Algebra with a View Towards Algebraic Geometry.  Springer, 1996.
  * [6] A.J. Engler and A. Prestel.  Valued Fields.  Springer Monographs in Mathematics. Springer Berlin Heidelberg, 2005.
  * [7] Joseph Flenner.  Relative decidability and definability in henselian valued fields.  Journal of Symbolic Logic, 76:1240–1260, 2011.
  * [8] M.D. Fried and M. Jarden.  Field Arithmetic.  Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics. 3rd Edition, Springer, 2008.
  * [9] Daniel Max Hoffmann, Piotr Kowalski, Chieu-Minh Tran, and Jinhe Ye.  Of model completeness and algebraic groups.  Preprint, available on https://arxiv.org/abs/2312.08988.
  * [10] Jaiung Jun.  Algebraic geometry over hyperrings.  Advances in Mathematics, 323(7):142–192, 2018.
  * [11] I. Kaplansky.  Infinite Abelian Groups.  Dover Books on Mathematics. Dover Publications, 2018.
  * [12] Marc Krasner.  Approximation des corps valués complets de caractéristique $p\neq 0$ par ceux de caractéristique 0.  Colloque d’algèbre supérieure, pages 129–206, 1957.
  * [13] Marc Krasner.  A class of hyperrings and hyperfields.  International Journal of Mathematics and Mathematical Sciences, 6(2):307–311, 1983.
  * [14] F.-V. Kuhlmann.  Quantifier elimination for henselian fields relative to additive and multiplicative congruences.  Israel Journal of Mathematics, 85:277–306, 1994.
  * [15] Junguk Lee.  Hyperfields, truncated DVRs and valued fields.  Journal of Number Theory, 212:40–71, 2020.
  * [16] Alessandro Linzi.  Notes on valuation theory for Krasner hyperfields.  Preprint, available on https://arxiv.org/pdf/2301.08639.
  * [17] Ch. G. Massouros.  Methods of constructing hyperfields.  International Journal of Mathematics and Mathematical Sciences, 8(4):725–728, 1985.
  * [18] Dan Segal and Katrin Tent.  Defining _R_ and _G(R)_.  J. Eur. Math. Soc, 25(8):3325–3358, 2023.
  * [19] Wanda Szmielew.  Elementary properties of Abelian groups.  Fundamenta Mathematicae, 41(2):203–271, 1955.
  * [20] Simon Thomas.  Classification theory of simple locally finite groups.  PhD thesis, University of London, 1983.
  * [21] Pierre Touchard.  Burden in henselian valued fields.  Annals of Pure and Applied Logic, 174(10), 2023.
  * [22] Pierre Touchard.  Stably embedded submodels of henselian valued fields.  Archive for Mathematical Logic, 63:279–315, 2023.