# Emergent Explainability: Adding a causal chain to neural network inference

Adam Perrett
The University of Manchester
Manchester, UK
<EMAIL_ADDRESS>
Corresponding author

###### Abstract

This position paper presents a theoretical framework for enhancing explainable
artificial intelligence (xAI) through emergent communication (EmCom), focusing
on creating a causal understanding of AI model outputs. We explore the novel
integration of EmCom into AI systems, offering a paradigm shift from
conventional associative relationships between inputs and outputs to a more
nuanced, causal interpretation. The framework aims to revolutionize how AI
processes are understood, making them more transparent and interpretable.
While the initial application of this model is demonstrated on synthetic data,
the implications of this research extend beyond these simple applications.
This general approach has the potential to redefine interactions with AI
across multiple domains, fostering trust and informed decision-making in
healthcare and in various sectors where AI’s decision-making processes are
critical. The paper discusses the theoretical underpinnings of this approach,
its potential broad applications, and its alignment with the growing need for
responsible and transparent AI systems in an increasingly digital world.

##  1 Introduction and background

###  1.1 Opportunities and challenges of explainability

Explainable artificial intelligence (xAI) remains one of the foremost
challenges in the field, crucial to equitable and fair application,
particularly in sensitive domains like healthcare. For examples, understanding
the impact of race in AI models is a common problemHuang et al. [2022].
Mitigating potential bias that exists within data and its effect on treatment
begins with understanding the AI model. The core issue lies in the inherent
opacity of advanced AI models, especially deep learning algorithms, which
operate as "black boxes". These complex models can make highly accurate
predictions or decisions, but understanding the causal chain leading to their
output is not straightforward. This lack of transparency raises concerns about
trustworthiness, accountability, and ethics, especially when AI is employed in
critical areas such as medical diagnostics, treatment planning, or patient
care management. This work addresses concerns of xAI with the goal of enabling
AI to be a vital and interpretable tool for clinicians and patients alike,
fostering trust and enabling informed decision-making. The novel framework
also has many potential applications beyond healthcare, with the potential to
reshape the way we interact with AI.

In the healthcare sector, xAI methods face significant challenges. To address
concerns of safety, equity and trustworthiness, an understanding of causality
is crucial. Simpler models like decision trees trade complexity for clarity,
essential where decision rationales are crucial. However, complex models using
tools like LIME Ribeiro et al. [2016] and SHAP Lundberg and Lee [2017] often
yield limited, sometimes misleading insights. Visualization techniques in deep
learning, such as attention maps, provide helpful interpretations but lack
precise correlations with neural network processes Erhan et al. [2009],
Simonyan et al. [2013]. Feature importance measures, while identifying key
decision factors, can suffer from bias and lack causal clarity Zhou et al.
[2016], Selvaraju et al. [2017]. These methods, enhancing reliability and
aiding compliance, struggle with a fundamental trade-off between accuracy and
explainability. This leads to technical complexities and the risk of
oversimplified interpretations, underscoring the need for more refined and
transparent explainability techniques in high-stakes healthcare AI
applications.

This work focuses on enhancing xAI in healthcare, a key area in need of growth
and equitable application. It aims to develop a novel and practical framework
for AI explainability, particularly in healthcare diagnostics and patient
care. The proposed research will imbue the AI model with a causal
communication channel that can be used for validation and transparency of
output. The expected outcomes include improved AI clarity and trust between
clinicians and patients, with the potential to inform policy in AI ethics.
This effort is geared towards contributing meaningful research and reshaping
the way we use AI as a tool.

Figure 1: The contextualiser network receives the task ID (e.g. task = is
there a dog) and passes a message to the actor network. The actor network
processes the message and an input to produce an output (e.g. is there a dog +
this image = yes/no). The actor is trained with supervised learning and the
contextualiser is trained with reinforcement learning, as errors cannot
propagate between networks.

###  1.2 Emergent communication

Emergent communication (EmCom) in AI Brandizzi [2023], a relatively nascent
field, presents unique challenges and opportunities in the context of
explainability. The principle behind EmCom is to have two or more agents
exchanging messages in order to solve a task, see Fig.1. Altering the task
set-up can alter the compositionality and generalisation of the generated
language Mu and Goodman [2021], Lowe et al. [2020]. Current research focuses
on investigating the generated language and comparing it to human language Li
and Bowling [2019], Lazaridou and Baroni [2020], Lowe et al. [2019]. The
research community sees the opportunity to apply this further and scale it up
Chaabouni et al. [2022], highlighting the growth possible in this fruitful
domain. A key shift this work aims to exploit is moving from message passing
being input dependant to task dependant. This raises the level of abstraction
of the message from a label to an instruction.

###  1.3 Emergent explainability

The aim of this research is to utilise the mechanisms of EmCom to make neural
network explainability causal. Current xAI methods reveal associative
relationships between inputs and outputs but are often limited to analysing
single input presentations and say little about causality. The proposed work
hinges on the following principles:

  * •

The messages used in EmCom can be made human interpretable (e.g. text or
image)

  * •

The message and the output share a causal relationship, as the task can only
be solved with the message

  * •

An appropriately generalised language facilitates: transfer learning, data
privatisation, distributed learning

##  2 Methodology

Experimental work focuses on synthetic data. Work has also been performed
using the MNIST dataset, although this is still currently under investigation.
Preliminary results will be discussed, with detailed analysis to follow.
Integration with human interpretable message passing is currently theoretical.

###  2.1 Experimental design

All possible truth tables for three inputs ($n=3$) act as the family of tasks
that are trained on. This gives $2^{2^{3}}=256$ possible truth tables and
$2^{3}*2^{2^{3}}=1024$ training examples. Each truth table is allocated a
random One Hot Encoding (OHE) to act as the task ID, this will be passed to
the contextualiser to generate a truth table specific message. A random OHE is
used to unsure there is as little possible information present in the task ID
and all communicated information must be learnt during training. The actor
will then receive the message and a set of 3 inputs.

###  2.2 Agent setup

Each agent has a separate contextualiser network and actor network. Both have
two hidden layers of 128 ReLU activation neurons, only difference between them
being their inputs, outputs, and initialisation. The conextualiser has as many
inputs as there are tasks, as it receives a OHE for the task ID. There is a
context layer of 32 ReLU neurons places after the final hidden layer. The
activation of these neurons forms the message that is passed. There is then a
single dense layer which samples from this to predict the expected reward of
the actor. The error between this and the actual actor reward forms the
reinforcement learning signal. The actor network has $3+32$ inputs (input size
+ context length) and 2 outputs (binary 0 and 1 for the truth table entry). It
is trained using supervised cross-entropy.

###  2.3 Communication protocol

Fig. 2 shows how the training data is distributed. Agents are allocated a
unique portion of tasks which it can act as a contextualiser for, $r_{c}$. It
can act as an actor for all these and an additional portion of the remaining
tasks, $r_{a}$. The remaining tasks will be unseen by the agent throughout
training. There is a truth table associated with each training example. For
each example an agent which can act as a contextualiser is randomly selected.
It is given as input the OHE associated with the task ID and produces the
associated context. An agent which can be an actor for this training example
is then selected and given the associated message and input.

###  2.4 Training

A batch size of 512 is used with an Adam optimiser using learning rate 0.001.
Cross-entropy loss between the target and actual output is used to train the
actor. Mean-squared error between the predicted and actual actor accuracy is
used to train the contextualiser using reinforcement learning. When the
contextualiser and actor are both the same agent, gradients are allowed to
flow back through the actor and into the communicated message.

Figure 2: An agent can act in three different capacities during training. It
can contextualise, meaning it takes the task ID and produces a message. It can
behave as the actor, meaning it uses the message and the input to produce the
target output during these training instances. It can be an actor for examples
it contextualises and ones other agents contextualise. There is then a subset
of data that is never seen by the agent, this is used to evaluate the
generalisation of the generated language.

###  2.5 Evaluation

The key evaluation metric is the performance of an agent on unseen truth
tables. The only way to achieve higher than random chance accuracy is for the
communicated messages to contain generalised information about the structure
of the family of tasks. This way, even though an agent has never performed the
task, it can understand the message and perform the task adequately.

###  2.6 Work in Progress - Human interpretability

This methodology is currently under development. The principle is to alter the
message passing between agents to be human interpretable. This can take the
form of an image related to the task, such as the features to look for e.g.
the task is looking for a dog so the communicated message contains a nose and
floppy ears. This can be injected into the convolutional kernels or
concatenated downstream. It is also possible that the message passing can be
human language. This would require the message to contain all relevant
information to complete the task.

##  3 Experiments and Results

###  3.1 Examining the effect of actor overlap

The variable $r_{a}$, controlling the ratio of examples in which an agent is
not a contextualiser but can be an actor is explored in Fig, 3. As the amount
of ’information sharing’ increases so does the classification accuracy on
unseen data. This displays that interaction between agents is key to
generating a generalised communication. This is most evident when there is no
sharing and the performance does not increase above random chance.

Figure 3: Classification accuracy on unseen data across all agents for n=3
truth table size. The number of example that are contextulised by an agent is
fixed and the parameter $r_{a}$, determining the number of examples that an
agent is an actor for but not a contextualiser, is investigated. It can be
seen that as the number of ’taught’ examples increased the generalisation of
the communication also increases, tending towards 100% testing accuracy.

##  4 Discussion and Limitations

This work forms a position paper displaying the potential for increasing
abstract message passing between agents compared to the current literature. It
has been displayed for communicating information about a family of tasks and
has been shown to generalise outside of tasks an agent is trained on.

There is still much work to do in imbuing this communication channel with
human interpretable messages but this is a stepping stone in that direction
and a signpost for future endeavours.

With the novel transition in this work of EmCom from communicating a label to
communicating task information, the language is a higher level of abstraction
and more focused on function. It has already been proven that languages can
emerge which generalise beyond training data Mu and Goodman [2021], ?. This
naturally leads to networks which can generalise across tasks, given the
appropriate context. This can allow training data to remain proprietary whilst
still sharing learnt information with another network. It can enable
distributed learning, as many agents can go off and learn different tasks and
return with their understanding. Obvious extensions to transfer learning exist
due to the task independent nature of the language. There is also reason to
believe that embedding such causal grounding in LLMs will help alleviate
problems such as hallucinations and mistakes with trivial request like simple
maths. This is because the framework will remove the sole dependence on the
statistical regularities of text and instead reposition the text as a medium
with which the task can be understood. When this condition is met the initial
contextualiser in the EmCom framework can be replaced with a human, and the
actor can then be harnessed as a generalised task solver.

## References

  * Huang et al. [2022] Jonathan Huang, Galal Galal, Mozziyar Etemadi, and Mahesh Vaidyanathan.  Evaluation and mitigation of racial bias in clinical machine learning models: Scoping review.  _JMIR Med Inform_ , 10(5):e36388, May 2022.  ISSN 2291-9694.  doi: 10.2196/36388.  URL https://medinform.jmir.org/2022/5/e36388.
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# Emotion-Aware Music Recommendation System: Enhancing User Experience Through
Real-Time Emotional Context

Tina Babua, Rekha R Naira and Geetha Aa CONTACT T. B. Author. Email:
<EMAIL_ADDRESS>aDepartment of Computer Science and Engineering, Alliance
University, Bengaluru, India;

###### Abstract

This study addresses the deficiency in conventional music recommendation
systems by focusing on the vital role of emotions in shaping users’ music
choices. These systems often disregard the emotional context, relying
predominantly on past listening behavior, and failing to consider the dynamic
and evolving nature of users’ emotional preferences. This gap leads to several
limitations. Users may receive recommendations that do not match their current
mood, which diminishes the quality of their music experience. Furthermore,
without accounting for emotions, the systems might overlook undiscovered or
lesser-known songs that have a profound emotional impact on users.

To combat these limitations, this research introduces an AI model that
incorporates emotional context into the song recommendation process. By
accurately detecting users’ real-time emotions, the model can generate
personalized song recommendations that align with the user’s emotional state.
This approach aims to enhance the user experience by offering music that
resonates with their current mood, elicits the desired emotions, and creates a
more immersive and meaningful listening experience. By considering emotional
context in the song recommendation process, the proposed model offers an
opportunity for a more personalized and emotionally resonant musical journey.

###### keywords:

emotion; classification; songs; recommendation; blockchain; CNN;

††articletype: ARTICLE TEMPLATE

##  1 Introduction

Music recommendation systems play a significant role in today’s digital
landscape, catering to users’ preferences and enhancing their listening
experience [1, 2]. Traditional recommendation algorithms predominantly rely on
users’ historical listening data and explicit preferences, overlooking the
essential aspect of emotions in music selection [3]. This absence limits the
capacity of recommendation systems to cater to the dynamic and evolving
emotional needs of users, impacting their overall music experience.

Recognition of emotions is reliant on the analysis of acoustic or visual
features’ dynamics. In Affective Emotion Recognition (AER), classification
methods incorporate both dynamic and static strategies [4]. Dynamic
classifiers, such as Hidden Markov Models (HMM) and Dynamic Bayesian Networks
(DBN), focus on the temporal nature of features [5]. Conversely, static
techniques like Support Vector Machines (SVM) process statistical functions
derived from extended data segments. The effectiveness of human-computer
interfaces, which emulate speech emotions, heavily depends on feature types
and the choice of classifiers. For instance, the study presented a unimodal
framework aimed at contextualizing short-term interactions within dyadic
contexts.

Recent studies have highlighted the crucial role emotions [6] play in
individuals’ music preferences. Users often seek songs that resonate with
their current emotional state, making emotional context a vital component in
song recommendations. The lack of emotional intelligence in existing systems
[7] can result in misaligned song suggestions, failing to capture the nuanced
and rapidly changing emotional needs of users.

In response to these limitations, a novel AI-based model [8] has been proposed
to detect users’ real-time emotions and generate music recommendations
tailored to their emotional state. Incorporating emotional context aims to
enrich users’ music experiences, providing songs that align with their moods,
elicit desired emotions, and offer a more immersive listening journey.

##  2 Proposed Methodology

The setup of an AI-based song recommendation system that incorporates the
Alchemy Blockchain API, CNN model [9], and Python involves a series of key
procedures. The architecture is shown in Figure 1. Initially, it encompasses
the collection and refinement of a dataset comprising songs annotated with
emotional labels or mood tags. Subsequently, the song lyrics undergo
preprocessing methods like tokenization, removal of stop words, and conversion
to numerical representations.

A CNN model is developed utilizing frameworks like Keras or TensorFlow,
incorporating convolutional [10], pooling, and fully connected layers for
extracting features and classifying emotions. This model is trained on the
prepared textual data, adjusting hyperparameters for optimal performance.
Integration with the Alchemy Blockchain API ensures secure storage and
retrieval of the recommendation system data, encompassing user preferences,
song metadata, and emotional tags [11].

Figure 1: Architecture of the System.

A user interface is created for users to input their emotions, linked to the
AI model for emotion classification [12]. A recommendation engine is
established to align user emotions with suitable songs, employing
collaborative or content-based filtering techniques. Emotional tags associated
with songs, along with the AI model’s predictions, contribute to enhancing
recommendation accuracy [13].

The system’s deployment and integration prioritize scalability, efficiency,
and real-time responsiveness. Comprehensive testing and assessment gather user
feedback for ongoing model and recommendation engine enhancements. Regular
updates and retraining of the AI model using new data are conducted to adapt
to evolving user preferences and emotions.

##  3 Experimental Results and Discussion

The utilization of an AI-based song recommendation system grounded on
emotional cues, coupled with blockchain technology, showed encouraging
outcomes. The CNN model’s ability to classify emotions was evaluated using key
metrics such as accuracy, precision, recall, and F1-score. This assessment
offered valuable insights into the model’s proficiency in predicting emotions
by analyzing song lyrics.

Figure 2: Happy mood detected successfully by the application.

The algorithm for personalized song recommendations effectively paired the
user’s identified emotions with music that suited their emotional condition.
This individualized method improved the music experience by offering users
song recommendations that were relevant to their emotional state.

Figure 3: Surprise mood detected successfully by the application.

Happy emotion was identified and the respective songs related to this was
recommended as given in the Figure 2. Similarly, Figure 3, Figure 4 shows the
recognition of surprise and disgust mood respectively. The neutral mood is
illustrated in Figure 5. The happy and sad mood combined is shown in Figure 6.

Figure 4: Disgusted mood detected successfully by the application. Figure 5:
Neutral mood detected successfully by the application.

Feedback from users and satisfaction surveys demonstrated affirmative
reactions to the AI-powered song recommendation system. Users valued the real-
time capture of their emotions and the system’s capability to propose songs
that aligned with their emotional condition.

Figure 6: Happy and sad mood detected successfully by the application.

The incorporation of the AI model for emotional-based song recommendations led
to enhanced user interaction and longer durations within the app. Users found
the personalized suggestions more attractive and were more inclined to explore
and find new songs that matched their emotional state. Consequently, user
retention within the app significantly increased.The amalgamation of the AI
model and the recommendation system uplifted the overall user journey. The
user-friendly interface and the ease of receiving emotional-based song
suggestions contributed to a more enjoyable and engaging music experience. The
visually pleasing neomorphic UI design elements further bolstered user
satisfaction.

(a) Implementation of Alchemy Blockchain.

(b) Number of Requests made per day.

Figure 7: Blockchain performance.

When compared to conventional recommendation techniques like collaborative and
content-based filtering, the AI model showcased superior recommendation
precision and customization. This illustrates the effectiveness of the
emotion-centric approach in delivering pertinent and personalized song
suggestions.

The system showed robustness and efficiency, effectively managing numerous
simultaneous user requests without sacrificing response speed or accuracy. Its
integration with external APIs, including the Spotify API, ensured a smooth
data flow and the delivery of high-caliber song recommendations.

The incorporation of blockchain technology extended advantages to the AI-
driven song recommendation system. The performance is illustrated in Figure 7
(a) and (b). It bolstered users’ data privacy and security by ensuring safe
and encrypted storage on the blockchain. The transparent aspects of blockchain
technology instilled confidence in the recommendation process. The
incentivization of user engagement via token reward systems led to a more
expansive and varied dataset for enhanced recommendations. Additionally, it
ensured copyright protection and attribution by immutably recording ownership
and usage rights. Moreover, the decentralized governance allowed users to
engage in decision-making processes, promoting fairness and inclusivity within
the recommendation system.

##  4 Conclusion

The merging of an AI model catering to emotion-based song recommendations with
blockchain technology represents a robust solution for enriching the music
listening journey. By precisely identifying emotions and providing
personalized song suggestions, users can relish a customized music assortment
that aligns with their emotional condition. This fusion of the AI model and
blockchain yields numerous substantial advantages and results. There are
extensive opportunities for further advancements and applications in the realm
of AI models for emotion-based song recommendations integrated with blockchain
technology in the future.

## References

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# On logically-geometric types of algebras

###### Abstract

The connection between classical model theoretical types (MT-types) and
logically-geometrical types (LG-types) introduced by B. Plotkin is considered.
It is proved that MT-types of two $n$-tuples in two universal algebras
coincide if and only if their LG-types coincide. An algebra $H$ is called
logically perfect if for every two $n$-tuples in $H$ whose types coincide, one
can be sent to another by means of an automorphism of this algebra. Some
sufficient condition for logically perfectness of free finitely generated
algebras is given which helps to prove that finitely generated free Abelian
groups, finitely generated free nilpotent groups and finitely generated free
semigroups are logically perfect. It is proved that if two Abelian groups have
the same type and one of them is finitely generated and free then these groups
are isomorphic.

G.Zhitomirski

Department of Mathematics, Bar-Ilan University,

52900, Ramat Gan, Israel

E-mail<EMAIL_ADDRESS>
##  1 Introduction

The ideas suggested and developed by B. Plotkin in the field of algebraic
logic seem to be very interesting and efficient. It turns out that the
geometrical notions and the geometrical intuition can be successfully applied
in studying algebras from arbitrary varieties. Such approach leads to so
called universal algebraic geometry and multi-sorted logical geometry.

The sketch of the ideas of universal algebraic geometry, problems and results
can be found for example in [6], [5], [7], [10], [11]. The notions of logical
geometry and obtained facts are presented in [12],[8], [9].

The purpose of this paper is to consider only one but important notion of
model theory, namely, the notion of type. The model theoretic notion of type
is well known [1]. Such a type is denoted in the paper by MT-type. MT-type is
related to one-sorted logics. On the other hand, the ideas of universal
logical geometry give rise to logically-geometric types (LG-types). This
notion is related to multi-sorted logic [12],[9]. Some of the problems
discussed in the literature are the following ones: how are connected
algebraically two $n$-tuples in an algebra whose types coincide and what we
can say about two algebras whose types coincide.

Let $\Theta$ be a variety of universal algebras of some signature and $W(X)$
denote the free $\Theta$-algebra over a set $X=\\{x_{1},x_{2},\dots,x_{n}\\}$.
In the universal algebraic geometry, the set $A^{n}$ of all $n$-tuples in a
$\Theta$-algebra $A$ is replaced by the set $\hom(W(X),A)$ which is called an
$n$-dimensional affine space and whose elements are called points. Since a
point $\mu\in\hom(W(X),A)$ is a map we can speak about its kernel. Along with
this usual kernel, so called logical kernel of $\mu$ is defined. The notion of
logical kernel of a point leads to the notion of LG-type of an algebra. All
notions mentioned above are defined in Section 2.

Although the two kinds of types mentioned above are related to different
languages we show that MT-types of two $n$-tuples coincide if and only if the
logical kernels of the corresponding points coincide (Theorem 3.1).

Then in Section 4 we consider so called logically perfect algebras. An algebra
$H$ is said to be logically perfect if for every its two $n$-tuples whose
types coincide there exists an automorphism of $H$ which sends one of these
tuples to another. A sufficient condition for logically perfectness of free
finitely generated algebras is given. The main result in this section is
Theorem 4.5.

The last Section 5 is devoted to algebras having the same type (isotyped
algebras). It is proved (Theorem 5.3) that if two Abelian groups have the same
type and one of them is finitely generated and free then these groups are
isomorphic.

The obtained results solve some problems set in [9].

Acknowledgments The author is pleased to thank B. Plotkin for useful
discussions and interesting suggestions.

##  2 Preliminaries

Throughout this paper, $\Theta$ is a variety of universal algebras of some
signature which determines the corresponding first-order language $L$ with
equality ”$\equiv$” and the infinite set $X^{0}=\\{x_{1},x_{2},\dots\\}$ of
variables. Let $W(X)$ denote the free $\Theta$-algebra generated by $X\subset
X^{0}$. We consider finite subsets $X\subset X^{0}$ only and follow the
conception suggested by B. Plotkin (see for example [6], [9], [8], [12]).

Let $\mathbb{M}$ be a $\Theta$-algebra with the domain $M$. Every $n$-tuple
$\bar{a}=(a_{1},\dots,a_{n})$ of elements of $M$ determines a homomorphism
$\mu:W(X)\to\mathbb{M}$ where $X=\\{x_{1},\dots,x_{n}\\}$, viz
$\mu(x_{i})=a_{i}$ for $i=1,\dots,n$. And vice versa, every such homomorphism
determines an $n$-tuple in $M$. Thus the set $M^{n}$ can be identified with
$\hom(W(X),\mathbb{M})$ which is called an affine space and whose elements are
called points. Considering the tuples in $M$ as points in the corresponding
affine space gives us new interesting opportunities.

First of all, the kernel of a point $\mu$ appears:
$Ker\mu=\\{(w,w^{\prime})|\mu(w)=\mu(w^{\prime})\\}$. It is useful to consider
equalities $w\equiv w^{\prime}$ instead of corresponding pairs in $W(X)$. Such
an approach leads to connections between sets of points and systems of
identities, that is, to something like to algebraic geometry for an universal
algebra. For details see papers cited above. In the present paper, we focus on
the notion of so called logical kernel of a point $\mu$: $LKer\mu$. We recall
the definition according to [9].

Let $\Gamma$ denote the set of all finite subsets of $X^{0}$. For every
$X\in\Gamma$, consider the signature $L_{X}=\\{\vee,\wedge,\neg,\exists x,x\in
X,M_{X},\\}$, where $M_{X}$ is the set of all equalities $w\equiv w^{\prime}$,
where $w,w^{\prime}\in W(X)$. By adding for every $X\in\Gamma$ symbols
$s=s^{XY}:W(X)\to W(Y)$, we obtain multi-sorted signature $L_{\Theta}$ . The
corresponding multi-sorted language is defined by induction on length and sort
of formulas.

######  Definition 2.1.

1\. Each equality $w\equiv w^{\prime}$ is a formula of the length zero and
sort $X$ if $w\equiv w^{\prime}\in M_{X}$.

2\. Let $u$ be a formula of the length $n$ and the sort $X$. Then the formulas
$\neg u$ and $\exists xu$ are the formulas of the same sort $X$ and the length
$(n+1)$.

3\. For the given $s:W(X)\to W(Y)$ we have the formula $s_{*}u$ with the
length $(n+1)$ and the sort $Y$.

4\. Let $u_{1}$ and $u_{2}$ be formulas of the same sort $X$ and the length
$n_{1}$ and $n_{2}$ accordingly. Then the formulas $(u_{1}\vee u_{2})$ and
$(u_{1}\wedge u_{2})$ have the length $(n_{1}+n_{2}+1)$ and the sort $X$.

The set of all formulas of the sort $X$ will be denote by $\Phi(X)$.

The value $Val^{X}_{H}(u)$ of a formula $u\in\Phi(X)$ in a $\Theta$-algebra
$H$ is defined according to the construction. Elements of $Val^{X}_{H}(u)$ are
points $\mu:W(X)\to H$.

######  Definition 2.2.

(1). $Val^{X}_{H}(w\equiv w^{\prime})=\\{\mu\mid\mu(w)=\mu(w^{\prime})\\}$.

(2). If $v=\exists xu$ and $u\in\Phi(X)$, then $\mu\in Val^{X}_{H}(v)$ if and
only if there exists a point $\nu:W(X)\to H$ such that $\nu$ coincides with
$\mu$ for all $y\in X$ besides $x$ and $\nu\in Val^{X}_{H}(u)$.

(3). If $u_{1},u_{2}\in\Phi(X)$ then $Val^{X}_{H}(u_{1}\vee
u_{2})=Val^{X}_{H}(u_{1})\cup Val^{X}_{H}(u_{2})$, $Val^{X}_{H}(u_{1}\wedge
u_{2})=Val^{X}_{H}(u_{1})\cap Val^{X}_{H}(u_{2})$.

(4). $Val^{X}_{H}(\neg u)=\hom(W(X),H)\setminus Val^{X}_{H}(u)$.

(5). Let $s:W(X)\to W(Y)$ be a homomorphism, $v\in\Phi(X)$ and $u=s_{*}v$.
Then $\mu\in Val^{Y}_{H}(u)$ if and only if $\mu\circ s\in Val^{X}_{H}(v)$.

######  Definition 2.3.

A formula $u\in\Phi(X)$ belongs to the logical kernel $LKer(\mu)$ of a point
$\mu:W(X)\to H$ if and only if $\mu\in Val^{X}_{H}(u)$.

The set $LKer(\mu)$ of formulas from $\Phi(X)$ is called logically-geometric
$X$-type of the point $\mu$ ($X$-$LG$-type).

######  Definition 2.4 ([12]).

The set $T$ of formulas from $\Phi(X)$ is called $X$-$LG$-type of the algebra
$H$, if there is a point $\mu:W(X)\to H$ such that $T=LKer(\mu)$.

Algebras $H_{1}$ and $H_{2}$ in $\Theta$ are called $LG$-isotyped, if for any
finite $X$, every $X$-type of the algebra $H_{1}$ is an $X$-type of the
algebra $H_{2}$ and vice versa.

######  Definition 2.5.

An algebra $H$ is called logically perfect if for every two points $\mu$ and
$\nu$ in $H$ having the same $X$-$LG$-type (that is, $LKer(\mu)=LKer(\nu)$)
there exists an automorphism $\varphi$ of $H$ such that $\mu=\varphi\circ\nu$,
that is, $\varphi$ transports $n$-tuple $(\nu(x_{1}),\dots,\nu(x_{n}))$ to
$n$-tuple $(\mu(x_{1}),\dots,\mu(x_{n}))$.

Now we recall the model-theoretical notion of type of an $n$-tuple $\bar{a}$.

######  Definition 2.6.

The type $tp^{\mathbb{M}}(\bar{a})$ consists of all formulas
$u(x_{1},\dots,x_{n})\in L$ with free variables $x_{1},\dots,x_{n}$ (all other
variables in this formula are bounded) such that $\mathbb{M}\models
u(a_{1},\dots,a_{n})$, that is, $u(x_{1},\dots,x_{n})$ is true under
interpretation which assigns $a_{i}$ to $x_{i}$.

Such a type will be called $MT$-type. It is worth to mention that we do not
consider types depending of parameters (the more general definition can be
found in [2]). The problem arises how two tuples are algebraically connected
if their $MT$-types coincide.

Two kinds of types defined above (MT- and LG-type) are sets of formulas in
different languages. We will prove below that the two points $\mu$ and $\nu$
have the same $X$-$LG$-type if and only if the $n$-tuples
$(\nu(x_{1}),\dots,\nu(x_{n}))$ and $(\mu(x_{1}),\dots,\mu(x_{n}))$ have the
same $MT$-type.

##  3 Relations between logical-geometrical types and model-theoretical types

######  Theorem 3.1.

Let $H_{1}$ and $H_{2}$ be $\Theta$-algebras. Let
$\bar{a}=(a_{1},\dots,a_{n})$ and $\bar{b}=(b_{1},\dots,b_{n})$ be $n$-tuples
in $H_{1}$ and $H_{2}$ respectively. Consider two corresponding points
$\nu:W(X)\to H_{1}$ and $\mu:W(X)\to H_{2}$ where $X=\\{x_{1},\dots,x_{n}\\}$,
$\nu(x_{i})=a_{i}$ and $\mu(x_{i})=b_{i}$, $i=1,\dots,n$. Then
$LKer(\nu)=LKer(\mu)$ if and only if
$tp^{H_{1}}(\bar{a})=tp^{H_{2}}(\bar{b})$.

###### Proof.

We will prove this statement by several steps.

######  Lemma 3.2.

$LKer(\nu)=LKer(\mu)\Rightarrow tp^{H_{1}}(\bar{a})=tp^{H_{2}}(\bar{b})$.

###### Proof.

Let $LKer(\nu)$=$LKer(\mu)$. Let $u\in tp^{H_{1}}(\bar{a})$. Under Definition
2.6, we have that $u=u(x_{1},\dots,x_{n},\;y_{1},\dots,y_{m})$ with listed
variables, where $x_{1},\dots,x_{n}$ have free occurrences only, all
$y_{1},\dots,y_{m}$ are bounded, and $H_{1}\models u(a_{1},\dots,a_{n})$.

On the other hand, according to Definition 2.1, $u\in\Phi(X\cup Y)$, where
$Y=\\{y_{1},\dots,y_{m}\\}$. Therefore for every homomorphism $\gamma:W(X\cup
Y)\to H_{1}$ such that $\gamma(x_{i})=a_{i}$,$i=1,\dots,n$ (values of
$\gamma(y_{j})$ do not influence), we have $\gamma\in Val^{X\cup
Y}_{H_{1}}(u)$ (see definitions (1)-(4) from 2.2). Consider arbitrary
homomorphism $s:W(X\cup Y)\to W(X)$ such that $s(x_{i})=x_{i}$, $i=1,\dots,n$
and construct the formula $v=s_{*}u\in\Phi(X)$. Since $\nu\circ
s(x_{i})=a_{i}$, $i=1,\dots,n$, we have $\nu\circ s\in Val^{X\cup
Y}_{H_{1}}(u)$. Under definition 2.2 (5), we obtain that $\nu\in
Val^{X}_{H_{1}}(v)$ and therefore $v\in LKer\nu$.

Since $LKer(\nu)$=$LKer(\mu)$, we have $v\in LKer\mu$, that is, $\mu\in
Val^{X}_{H_{2}}(v)$ which implies that $\mu\circ s\in Val^{X\cup
Y}_{H_{2}}(u)$. Let $\delta:W(X\cup Y)\to H_{2}$ be an an arbitrary
homomorphism such that $\delta(x_{i})=b_{i}$ for all $i=1,\dots,n$. Since
$\mu\circ s:W(X\cup Y)\to H_{2}$, $\mu\circ s(x_{i})=b_{i}$ for all
$i=1,\dots,n$ and the variables from $Y$ are bounded in $u$ , we obtain that
the values of the formula $u$ under interpretations $\delta$ and $\mu\circ s$
coincide. Therefore $H_{2}\models u(b_{1},\dots,b_{n})$, that is, $u\in
tp^{H_{2}}(\bar{b})$. Consequently $tp^{H_{1}}(\bar{a})\subseteq
tp^{H_{2}}(\bar{b})$. The inverse inclusion is also true by symmetry. ∎

Now we assign to every formula $u\in\Phi(X),\;X\in\Gamma,$ a formula
$\tilde{u}$ in the one-sorted first order language, that is, a formula which
does not contain symbols $s_{*}$. Let $\tilde{X}^{0}$ be a copy of $X^{0}$
such that to every variable $x\in X^{0}$ the variable
$\tilde{x}\in\tilde{X}^{0}$ corresponds one to one. Consider the first-order
language $L$ associated with the variety $\Theta$ with set
$X^{0}\cup\tilde{X}^{0}$ of variables using variables from $X^{0}$ for free
variables and variables from $\tilde{X}^{0}$ for bounded ones only.

We construct the formula $\tilde{u}$ for every formula
$u\in\Phi(X),\;X\in\Gamma$, inductively.

1\. If $u$ is $w\equiv w^{\prime}$ then $\tilde{u}=u$.

2\. If $u$ is $\neg v,(u_{1}\vee u_{2})$ or $(u_{1}\wedge u_{2})$ then
$\tilde{u}=\neg\tilde{v},\;(\tilde{u_{1}}\vee\tilde{u_{2}})$ or
$(\tilde{u_{1}}\wedge\tilde{u_{2}})$ respectively.

3\. If $u=\exists xv$ and $x\in X$ then
$\tilde{u}=\exists\tilde{x}\tilde{v}|^{x}_{\tilde{x}}$, where
$\tilde{v}|^{x}_{\tilde{x}}$ denotes the formula in $L$ which is obtained by
replacing of all occurrences of the variable $x$ in $\tilde{v}$ by
$\tilde{x}$.

4\. Let $Y=\\{y_{1},\dots,y_{m}\\}\in\Gamma$ and $s:W(Y)\to W(X)$ be a
homomorphism, $v\in\Phi(Y)$ and $u=s_{*}v$. Then
$\tilde{u}=\tilde{v}|^{y_{1}}_{s(y_{1})},\dots,^{y_{m}}_{s(y_{m})}$. Notice
that all occurrences of elements from $X$ and $Y$ in $\tilde{v}$ can be free
only.

######  Lemma 3.3.

For every point $\mu:W(X)\to H$ and every $u\in\Phi(X)$

$u\in LKer(\mu)\Leftrightarrow\tilde{u}\in tp^{H}(\bar{a}),$

where $\bar{a}=(\mu(x_{1}),\dots,\mu(x_{n}))$, $X=\\{x_{1},\dots,x_{n}\\}$.

###### Proof.

We will prove this statement by induction according to the construction of
formulas of sort $X$.

1\. Let $u$ be $w\equiv w^{\prime}$. Under definition, $u\in LKer(\mu)$ means
that $\mu(w)=\mu(w^{\prime})$. In the considered case, $\tilde{u}=u$ and we
obtain that $u\in LKer(\mu)$ is equal to
$H\models\tilde{u}(a_{1},\dots,a_{n})$, that is, to $\tilde{u}\in
tp^{H}(\bar{a})$.

2\. For $u=\neg v,(u_{1}\vee u_{2}$) or $(u_{1}\wedge u_{2})$ our statement is
obviously true.

3\. Let $u=\exists xv$ , where $x\in X$. Assume that our statement is true for
$v$. The fact $u\in LKer(\mu)$ means that there exists a point $\nu:W(X)\to H$
which coincides with $\mu$ for all $y\in X$ besides $x$ and such that $\nu\in
Val^{X}_{H}(v)$. Under assumption, $\nu\in Val^{X}_{H}(v)$ is equal to
$\tilde{v}\in tp^{H}(\bar{b})$ where $\bar{b}=(\nu(x_{1}),\dots,\nu(x_{n}))$.
Since $\tilde{u}=\exists\tilde{x}\tilde{v}|^{x}_{\tilde{x}}$, we obtain that
$u\in LKer(\mu)$ is equal to $\tilde{u}\in tp^{H}(\bar{a})$ where
$\bar{a}=(\mu(x_{1}),\dots,\mu(x_{n}))$. Notice that $\tilde{u}$ does not
contain $x$.

4\. Let $Y=\\{y_{1},\dots,y_{m}\\}$, $s:W(Y)\to W(X)$ be a homomorphism,
$v\in\Phi(Y)$, and $u=s_{*}v$. Assume that our statement is true for $v$. This
means that $v\in LKer(\mu\circ s)$ is equal to $\tilde{v}\in tp^{H}(\bar{b})$,
where $\bar{b}=\mu\circ s(\bar{y})$. Further, $v\in LKer(\mu\circ s)$ is equal
to $u\in LKer(\mu)$ and $\tilde{v}\in tp^{H}(\bar{b})$ is equal to
$\tilde{u}\in tp^{H}(\bar{a})$ because
$\tilde{u}=\tilde{v}|^{y_{1}}_{s(y_{1})},\dots,^{y_{m}}_{s(y_{m})}$ according
to the definition, and hence $H\models\tilde{u}(a_{1},\dots,a_{n})$ is the
same that $H\models\tilde{v}(b_{1},\dots,b_{m})$. Thus our statement is true
for $u$ too.

∎

######  Lemma 3.4.

$tp^{H_{1}}(\bar{a})=tp^{H_{2}}(\bar{b})\Rightarrow LKer(\nu)=LKer(\mu)$

###### Proof.

Let $tp^{H_{1}}(\bar{a})=tp^{H_{2}}(\bar{b})$. Let $u\in\Phi(X)$ and $u\in
LKer\nu$. Then according to Lemma 3.3, $\tilde{u}\in tp^{H_{1}}(\bar{a})$.
Consequently $\tilde{u}\in tp^{H_{2}}(\bar{b})$ and therefore $u\in LKer\mu$
according to the same Lemma. ∎

In virtue of Lemmas 3.2 and 3.4, Theorem 3.1 is proved. ∎

##  4 Logically perfect algebras

The purpose of this section is to present some results concerning logically
perfect algebras. Some authors call an algebra $H$ homogeneous if every
automorphism between two finitely generated subalgebras of $H$ can be extended
to an automorphism of $H$. It is easy to see that every homogeneous algebra is
logically perfect [8].

It is obvious that every finite dimensional linear space $V$ is a homogeneous
algebra, and therefore $V$ is logically perfect. On the other hand, it is easy
to see that free finitely generated semigroups and free finitely generated
Abelian groups are not homogeneous, nevertheless we will show below that all
of them are logically perfect. Thus the homogeneity is not a necessary
condition for an algebra to be logically perfect. There is a logical condition
equivalent to homogeneity obtained by the author. This condition is cited in
[8] and called there strictly logically perfectness. The following
generalization of homogeneity will be useful.

######  Definition 4.1.

An algebra $H$ is called weakly homogeneous if for every isomorphism
$\varphi:A\to B$ between two its finitely generated subalgebras $A$ and $B$,
the following condition is satisfied: if $\varphi$ itself and its inverse map
$\varphi^{-1}:B\to A$ both can be extended to endomorphisms of $H$ then
$\varphi$ can be extended to an automorphism of $H$.

######  Theorem 4.2.

Every weakly homogeneous finitely generated free algebra is logically perfect.

###### Proof.

Let $H$ be weakly homogeneous and $e_{1},...,e_{n}$ be free generators of $H$.
Let $X=\\{x_{1},...,x_{k}\\}$. Consider two points $\nu,\mu:W(X)\to H$ and
suppose that $LKer\nu=LKer\mu$. Let $\nu(x_{i})=a_{i}$ and $\mu(x_{i})=b_{i}$
for all $i=1,...,k$. Take $Y=\\{y_{1},...,y_{n}\\}$, such that $X\bigcap
Y=\emptyset$, and define a homomorphism $\gamma:W(Y)\to H$ by the values:
$\gamma(y_{i})=e_{i},\;i=1,...,n$. Let $w_{1},...,w_{k}\in W(Y)$ be any $k$
words such that $a_{i}=\gamma(w_{i}),\;i=1,...,k$.

Consider a formula $u$ of sort $X$ of the kind $u=s_{*}(v)$ where

$v=(\exists y_{1})...(\exists y_{n})(x_{1}\equiv w_{1}\wedge...\wedge
x_{k}\equiv w_{k})$

and $s:W(X\bigcup Y)\to W(X)$ defined by
$s(x_{i})=x_{i},s(y_{1})=...=s(y_{n})=x_{1}$.

It is obvious that $\nu\in Val_{H}(u)$. Thus under assumption, $\mu\in
Val_{H}(u)$ and therefore $\mu\circ s\in Val_{H}(v)$. The last one means that
there exists a homomorphism $\delta:W(Y)\to H$ such that
$b_{i}=\delta(w_{i}),\;i=1,...,k$. Define an endomorphism $\sigma$ of $H$
setting $\sigma(e_{i})=\delta(y_{i}),\;i=1,...,n$, that is,
$\sigma\circ\gamma=\delta$. We have
$\sigma(a_{i})=\sigma(\gamma(w_{i}))=\delta(w_{i})=b_{i}$ for $i=1,...,k$.
Hence $\sigma$ determines a homomorphism $\varphi$ of the subalgebra $A$
generated by $a_{1},...,a_{k}$ on the subalgebra $B$ generated by
$b_{1},...,b_{k}$.

Similarly, we can define an endomorphism $\tau$ of $H$ such that
$\tau(b_{i})=a_{i}$ for $i=1,...k$. Consequently
$\sigma\circ\tau(b_{i})=b_{i}$ and $\tau\circ\sigma(a_{i})=a_{i}$ which means
that the restriction $\varphi$ of $\sigma$ to $A$ is an isomorphism of $A$ on
$B$ and $\varphi^{-1}$ is a restriction of $\tau$. Since $H$ is weakly
homogeneous, $\varphi$ can be extended up to automorphism $\tilde{\varphi}$ of
$H$ for which we have $\tilde{\varphi}\circ\nu=\mu$. ∎

######  Lemma 4.3.

Finitely generated free Abelian groups and finitely generated free nilpotent
groups are weakly homogeneous.

###### Proof.

1. We start with considering Abelian groups. Let $G$ and $F$ be free Abelian groups of the same rank $n$. Let $A$ and $B$ be two subgroups of $G$ and $F$ respectively which are isomorphic by means of an isomorphism $\varphi:A\to B$. We will prove that if $\varphi$ and $\varphi^{-1}$ both can be extended up to homomorphisms $\sigma:G\to F$ and $\tau:F\to G$ respectively, then $\varphi$ can be extended up to an isomorphism of $G$ onto $F$.

It is known ([4], Theorem 3.5) that there exists a base $g_{1},...,g_{n}$ of
$G$ and a base $a_{1},...,a_{k}$ of $A$ such that $a_{i}=p_{i}g_{i}$ for
$1\leq i\leq k$ where $p_{1},...,p_{k}$ are integers and every $p_{i+1}$ is
divisible by $p_{i}$ for $1\leq i\leq k-1$ . Exactly in the same way, there
exists a base $f_{1},...,f_{n}$ of $F$ and a base $b_{1},...,b_{k}$ of $B$
such that $b_{i}=q_{i}f_{i}$ for $1\leq i\leq k$ and every integer $q_{i+1}$
is divisible by the integer $q_{i}$ for $1\leq i\leq k-1$.

Let $\sigma(g_{i})=\sum_{j=1}^{n}s^{j}_{i}f_{j}$ and
$\tau(f_{i})=\sum_{j=1}^{n}t^{j}_{i}g_{j}$. We obtain two integer matrices of
order $n$: $S=||s^{j}_{i}||$ and $T=||t^{j}_{i}||$. Since $\varphi:A\to B$ is
an isomorphism, $\varphi$ provides an invertible integer matrix
$||a^{j}_{i}||$ of order $k$, where
$\varphi(a_{i})=\sum_{j=1}^{k}a^{j}_{i}b_{j}$. Let $||b^{j}_{i}||$ be its
inverse matrix: $\varphi^{-1}(b_{i})=\sum_{j=1}^{k}b^{j}_{i}a_{j}$.

Since $\sigma(a_{i})=\varphi(a_{i})$, we obtain
$p_{i}\sigma(g_{i})=\sum_{j=1}^{k}a^{j}_{i}b_{j}=\sum_{j=1}^{k}a^{j}_{i}q_{j}f_{j}$
for $1\leq i\leq k$. Thus for all $1\leq i\leq k$ we have
$p_{i}\sum_{j=1}^{n}s^{j}_{i}f_{j}=\sum_{j=1}^{k}a^{j}_{i}q_{j}f_{j}$. This
implies that $p_{i}s^{j}_{i}=a^{j}_{i}q_{j}$ for $1\leq i,j\leq k$ and
$p_{i}s^{j}_{i}=0$ for $1\leq i\leq k$, $k+1\leq j\leq n$. In view of the
definitions of $p_{i},q_{i}$, we have $p_{1}=q_{1}$ and $s^{j}_{i}=0$ for all
$1\leq i\leq k$ and $k+1\leq j\leq n$.

By duality, we obtain $q_{i}t^{j}_{i}=b^{j}_{i}p_{j}$ for $1\leq i,j\leq k$
and $t^{j}_{i}=0$ for all $1\leq i\leq k$ and $k+1\leq j\leq n$. Therefore we
obtain for all $1\leq i,j\leq k$ :

$\sum_{l=1}^{k}s^{j}_{l}t^{l}_{i}=\sum_{l=1}^{k}\frac{a^{j}_{l}q_{j}}{p_{l}}\frac{b^{l}_{i}p_{l}}{q_{i}}=\sum_{l=1}^{k}\frac{q_{j}}{q_{i}}a^{j}_{l}{b^{l}_{i}}=\begin{cases}1,&\text{if
$i=j$;}\\\ 0,&\text{if $i\not=j$.}\end{cases}$ (1)

Consider the left corner $k$-th minor $M$ of the matrix $S$, that is, the
determinant of the matrix $||s^{j}_{i}||_{1\leq i,j\leq k}$. According to (1)
$M=1$ or $M=-1$. Define map $\tilde{\varphi}:G\to F$ setting

$\tilde{\varphi}(g_{i})=\begin{cases}\sigma(g_{i}),&\text{if $i\leq k$;}\\\
f_{i},&\text{if $k+1\leq i\leq n$.}\end{cases}$ (2)

The matrix $V$ of this map is

$V=\left(\begin{matrix}s^{1}_{1}&...&s^{1}_{k}&0&...&0\\\
...&...&...&0&...&0\\\ s^{k}_{1}&...&s^{k}_{k}&0&...&0\\\ 0&...&0&1&...&0\\\
0&...&0&0&1...&0\\\ 0&...&0&0&...&1\end{matrix}\right)$

We see that $DetV=M=\pm 1$ and therefore $\tilde{\varphi}$ is an isomorphism.
By construction,
$\tilde{\varphi}(a_{i})=m_{i}\tilde{\varphi}(g_{i})=m_{i}\sigma(g_{i})=\sigma(a_{i})=\varphi(a_{i})$
for all $i\leq k$. Consequently $\tilde{\varphi}$ extends $\varphi$.

2. Now let $H$ be a finitely generated free nilpotent group of class $c>1$ and rank $n$. Let $A$ and $B$ be two subgroups of $H$ which are isomorphic by means of an isomorphism $\varphi:A\to B$. Let $\varphi$ and $\varphi^{-1}$ both can be extended up to endomorphisms $\sigma$ and $\tau$ of $H$ respectively.

The quotient group $G=H/H^{\prime}$ is a free Abelian group of the same rank
$n$. Let $\eta:H\to G$ be the corresponding epimorphism. Then
$\bar{A}=\eta(A)$ and $\bar{B}=\eta(B)$ are isomorphic subgroups of $G$ under
isomorphism $\bar{\varphi}=\eta\circ\varphi\circ\eta^{-1}$. This isomorphism
is contained in the endomorphism $\bar{\sigma}=\eta\circ\sigma\circ\eta^{-1}$
and the inverse isomorphism $\bar{\varphi}^{-1}$ is contained in the
endomorphism $\bar{\tau}=\eta\circ\tau\circ\eta^{-1}$. Thus we can apply the
fact proved above in the point 1, that is, $\bar{\varphi}$ can be extended up
to automorphism $\bar{\Phi}$ of $G$.

Consider this extension in details. A base $g_{1},...,g_{n}$ of Abelian group
$G$ and a base $\bar{a}_{1},...,\bar{a}_{k}$ of its subgroup $\bar{A}$ are
chosen such that $\bar{a}_{i}=g_{i}^{p_{i}}$ for $1\leq i\leq k$ (now we use
the multiplicative notation). The automorphism $\bar{\Phi}$ of $G$ extending
$\bar{\varphi}$ is constructed in such a way that
$\bar{\Phi}(g_{i})=\bar{\sigma}(g_{i})$ for $i\leq k$. The elements
$f_{i}=\bar{\Phi}(g_{i})$ for $i=1,\dots,n$ form a base of $G$ in which first
$k$ elements are equal to corresponding $\bar{\sigma}(g_{i})$.

It is known from the theory of nilpotent groups (see for example [3]) that a
system $h_{1},\dots,h_{n}$ of elements of $H$ is a system of free generators
of some free nilpotent subgroup of the same class if and only if the the
system $\eta(h_{1}),\dots,\eta(h_{n})$ is linear independent in
$G=H/H^{\prime}$. So if $\eta(h_{1}),\dots,\eta(h_{n})$ is a base of $G$ then
$h_{1},\dots,h_{n}$ is a base of a free nilpotent subgroup $H_{0}$ of $G$.
Since $\eta(H_{0})=G$, we have $H_{0}H^{\prime}=H$. The last one implies that
$H_{0}=H$. We obtain that if $\eta(h_{1}),\dots,\eta(h_{n})$ is a base of $G$
then $h_{1},\dots,h_{n}$ is a base of $H$. Below we apply this property of
finitely generated free nilpotent groups.

There exist bases $h_{1},\dots,h_{n}$ and $u_{1},\dots,u_{n}$ of $H$ such that
$\eta(h_{i})=g_{i}$ and $\eta(u_{i})=f_{i}$ for $1\leq i\leq n$. Of course we
can chose $u_{i}=\sigma(h_{i})$ for $1\leq i\leq k$ because
$\eta(\sigma(h_{i}))=\bar{\sigma}(g_{i})=f_{i}$ for $1\leq i\leq k$.

Now we define an automorphism $\Phi$ of $H$ setting $\Phi(h_{i})=u_{i}$ for
$1\leq i\leq n$. On the other hand, elements $h_{i}^{p_{i}}$ $1\leq i\leq k$
form a base of the free nilpotent subgroup $AH^{\prime}$ because
$\eta(h_{i}^{p_{i}})=g_{i}^{p_{i}}=\bar{a_{i}}$. We have
$\Phi(h_{i}^{p_{i}})=(\Phi(h_{i}))^{p_{i}}=u_{i}^{p_{i}}=(\sigma(h_{i}))^{p_{i}}=\sigma(h_{i}^{p_{i}})$.
Thus $\Phi$ coincides with $\sigma$ on the subgroup $AH^{\prime}$. Since
$\sigma$ contains $\varphi$ which is defined on $A\subset AH^{\prime}$, $\Phi$
is an extension of $\varphi$. ∎

######  Lemma 4.4.

Every finitely generated free semigroup is weakly homogeneous.

###### Proof.

Let $S$ be a free semigroup with the set $X=\\{x_{1},\dots,,x_{k}\\}$ of free
generators . Let $\varphi:A\to B$ be an automorphism between two subsemigroups
$A$ and $B$ of $S$, where $A$ and $B$ are generated by elements
$a_{1},\dots,a_{n}$ and $b_{1},\dots,b_{n}$ respectively. We may assume that
$\varphi(a_{i})=b_{i}$ for $1\leq i\leq n$.

Suppose that there exist two endomorphisms $\sigma$ and $\tau$ first of which
extends $\varphi$ and the second one extends $\varphi^{-1}$. Thus
$\sigma(a_{i})=b_{i}$ and $\tau(b_{i})=a_{i}$ Denote by $|w|$ the length of
the word $w$ in alphabet $X$. Since $|\sigma(w)|\geq|w|$ and
$|\tau(w)|\geq|w|$ for every $w\in S$, we obtain that $|a_{i}|=|b_{i}|$. Let
$y_{1},\dots,y_{p}$ be the list of all variables from $X$ which occur in
$a_{1},\dots,a_{n}$ and $z_{1},\dots,z_{q}$ be the analogical list of all
variables which occur in $b_{1},\dots,b_{n}$. It is obvious that
$|\sigma(y_{i})|=1$ for all $1\leq i\leq p$ and $|\tau(z_{i})|=1$ for all
$1\leq i\leq q$. Therefore we have that
$\sigma(y_{i})\in\\{z_{1},\dots,z_{q}\\}$ and
$\tau(z_{i})\in\\{y_{1},\dots,y_{p}\\}$.

Since $\tau(\sigma(a_{i}))=a_{i}$ and $\sigma(\tau(b_{i}))=b_{i}$ for $1\leq
i\leq n$, we have that the restrictions of $\sigma$ and $\tau$ to variables
$y_{1},\dots,y_{p}$ and $z_{1},\dots,z_{q}$ respectively are mutually inverse
maps. Thus $p=q$ and $\sigma$ and $\tau$ induce two mutually inverse partial
one-to-one transformations of $X$. Let $\alpha$ be a bijection of
$X\setminus\\{y_{1},\dots,y_{p}\\}$ on $X\setminus\\{z_{1},\dots,z_{p}\\}$.
Setting $\tilde{\varphi}(y_{i})=\sigma(y_{i})$ for $1\leq i\leq p$ and
$\tilde{\varphi}(x)=\alpha(x)$ for all other variables from $X$, we obtain the
automorphism $\tilde{\varphi}$ of $S$ which extends $\varphi$. ∎

Lemmas 4.3, 4.4 and 4.2 give us the following result:

######  Theorem 4.5.

Finitely generated free Abelian groups, finitely generated free nilpotent
groups of any class and finitely generated semigroups are logically perfect.

The method which has been used to prove the theorem above can not be applied
to non-Abelian finitely generated free groups.

######  Proposition 4.6.

Free groups of rank 2 are not weakly homogeneous.

###### Proof.

Consider the free group $\mathbb{F}_{2}$ of rank 2 free generated by
$x_{1},x_{2}$. Let $a=x_{1}^{2}x_{2}x_{1}^{-1}x_{2}$ and $b=x_{1}x_{2}$.
Define endomorphisms $\sigma$ and $\tau$ setting
$\sigma(x_{1})=x_{1}x_{2},\;\sigma(x_{2})=1$ and
$\tau(x_{1})=x_{1}^{2}x_{2},\;\tau(x_{2})=x_{1}^{-1}x_{2}$. We see that
$\sigma(a)=b$ and $\tau(b)=a$. Thus $\sigma$ induces an isomorphism of
$\varphi:\langle a\rangle\to\langle b\rangle$ and $\tau$ induces the inverse
isomorphism $\varphi^{-1}$.

Suppose that there exists an automorphism $\tilde{\varphi}$ of
$\mathbb{F}_{2}$ which sends $a$ to $b$. Let $\tilde{\varphi}(x_{1})=w_{1}$ ,
$\tilde{\varphi}(x_{2})=w_{2}$, where $w_{1}$,$w_{2}$ are words in symbols
$x_{1},x_{2}$. Thus we have a relation in our free group: $x_{1}x_{2}\equiv
w_{1}^{2}w_{2}w_{1}^{-1}w_{2}$. (*)

This relation must be an identity in the group variety. Let $l_{1},l_{2}$ be
the sums of all exponents of $x_{1},x_{2}$ incoming in $w_{1}$ and
$m_{1},m_{2}$ the sums of all exponents of $x_{1},x_{2}$ incoming in $w_{2}$
respectively. It is obvious that $l_{1}+2m_{1}=l_{2}+2m_{2}=1$. Thus
$l_{1},l_{2}$ must be odd numbers.

Consider the group $S_{3}$ of all permutations of the set $\\{1,2,3\\}$ . This
group is a homomorphic image of $\mathbb{F}_{2}$ under the map $\gamma$ which
maps $x_{1}$ to $(213)$ and $x_{2}$ to $(132)$. Since
$\gamma(x_{1}^{2})=\gamma(x_{2}^{2})=(123),\;\gamma(x_{1}x_{2})=(312),\;\gamma(x_{2}x_{1})=(231),\;\gamma(x_{1}x_{2}x_{1})=\gamma(x_{2}x_{1}x_{2})=(321),\;\gamma((x_{1}x_{2})^{2})=\gamma(x_{2}x_{1}),\;\gamma((x_{2}x_{1})^{2})=\gamma(x_{1}x_{2})$,
we obtain that the following equalities are satisfied in $S_{3}$: $w_{1}\equiv
x_{1}x_{2}$ or $w_{1}\equiv x_{2}x_{1}$. For $w_{2}$ we have variants:
$w_{2}\equiv 1,x_{1},x_{2},x_{1}x_{2},x_{2}x_{1},x_{1}x_{2}x_{1}$. Since
$w_{1},w_{2}$ generate $\mathbb{F}_{2}$, their images generate $S_{3}$.
Therefore we have only three variants for $w_{2}$: $w_{2}\equiv
x_{1},x_{2},x_{1}x_{2}x_{1}$. Directly calculations show that in all mentioned
cases $\gamma(w_{1}^{2}w_{2}w_{1}^{-1}w_{2})=(123)$ which contradicts to the
identity (*).

Consequently there is no automorphism of $\mathbb{F}_{2}$ sending $a$ to $b$.
∎

Nevertheless all free finitely generated non-Abelian free groups are logically
perfect. This fact is proved in [2] in view of Theorem 3.1.

##  5 Isotyped algebras

We consider the following problem: in what cases isotyped algebras are
necessarily isomorphic. At first, we generalize the result obtained in [12],
Theorem 3.11.

######  Theorem 5.1.

If two algebras $H_{1}$ and $H_{2}$ from the same variety $\Theta$ are
isotyped then for every finitely generated subalgebra $A$ of $H_{1}$ there
exists a subalgebra $B$ of $H_{2}$ isomorphic to $A$, and if $A$ is a proper
subalgebra then $B$ can be chosen as a proper subalgebra too.

###### Proof.

Let $H_{1}$ and $H_{2}$ be isotyped $\Theta$-algebras. Let $A=\langle
a_{1},\dots,a_{n}\rangle$ where $a_{1},\dots,a_{n}$ are different elements in
$H_{1}$. Consider the free $\Theta$-algebra $W(X)$, where
$X=\\{x_{1},\dots,x_{n}\\}$. Let $\nu\in\hom(W(X),H_{1})$ defined by
$\nu(x_{i})=a_{i}$ for $1\leqq i\leqq n$. Since $H_{1}$ and $H_{2}$ are
isotyped there exists a point $\mu\in\hom(W(X),H_{2})$ such that
$LKer\nu=LKer\mu$. We obtain a subalgebra
$B=\langle\mu(a_{1}),\dots,\mu(a_{n})\rangle$ of $H_{2}$ and $B=\mu(W(X))$.
Since $Ker\nu=Ker\mu$, algebras $A$ and $B$ are isomorphic.

Let now $A$ be a proper subalgebra of $H_{1}$ and let $a_{n+1}\in
H_{1}\setminus A$. Add to $X$ a new variable $x_{n+1}\not\in X$ and consider a
new point $\nu:W(X\cup\\{x_{n+1}\\})\to H_{1}$ setting $\nu(x_{i})=a_{i}$ for
all $1\leqq i\leqq n+1$. For every $w\in W(X)$ consider the following formula
$v_{w}\in\Phi(X\cup\\{x_{n+1}\\})$:

$v_{w}=\neg(x_{n+1}\equiv w).$

Under condition that $H_{1}$ and $H_{2}$ are isotyped, there exists a point
$\mu\in\hom(W(X\cup\\{x_{n+1}\\}),H_{2})$ such that $LKer\nu=LKer\mu$. Since
$LKer\nu\cap M_{X}=LKer\mu\cap M_{X}$, the subalgebra $B$ generated by
$\mu(x_{1}),\dots,\mu(x_{n})$ is isomorphic to $A$. On the other hand, it is
obvious that $v_{w}\in LKer\nu$ and hence $v_{w}\in LKer\mu$ for every $w\in
W(X)$. The last one means that $\mu(x_{n+1})$ does not belong to $B$, that is,
$B$ is a proper subalgebra of $H_{2}$.

∎

######  Corollary 5.2.

Let a finitely generated algebra $H$ contain no proper subalgebra isomorphic
to $H$. Then every algebra $G$ isotyped to $H$ is isomorphic to $H$.

###### Proof.

Let $H$ and $G$ be isotyped algebras. Since $H$ is finitely generated, there
exists a subalgebra $B$ of $G$ isomorphic to $H$. If $B$ is a proper
subalgebra of $G$ then $H$ contains a proper subalgebra $A$ which is
isomorphic to $B$ and therefore $A$ is isomorphic to $H$ but this is
impossible according to the hypotheses. Thus $B=G$. ∎

We can apply this result to finitely dimensional linear spaces but it is not
the case for finitely generated free Abelian groups. However the next result
can be obtained using Theorem 5.1.

######  Theorem 5.3.

If two Abelian groups are isotyped and one of them is free and finitely
generated then they are isomorphic.

###### Proof.

Let $H$ and $G$ be isotyped Abelian groups and $H$ be free of rank $n$. Then
every finitely generated subgroup of $G$ is isomorphic to a subgroup of $H$.
Therefore every finitely generated subgroup of $G$ is free of a rank $k\leq
n$. This means that every $n+1$ elements of $G$ are linearly dependent. On the
other hand, $H$ is isomorphic to a subgroup $B$ of $G$. Let
$g_{1},\dots,g_{n}$ is a base of $B$. These elements form a maximal linearly
independent system in $G$. We obtain that rank of $G$ is equal to $n$.

It remains to show that $G$ is finitely generated. Let $h_{1},\dots,h_{n}$ be
a base of $H$. Consider the following countable set of formulas
$u_{(q_{1},\dots,q_{n})}(x_{1},\dots,x_{n})$, indexed by $n$-tuples
$(q_{1},\dots,q_{n})$ of integers, which not all are equal to zero and
formulas $v_{(q_{1},\dots,q_{n},q)}(x_{1},\dots,x_{n})$, indexed by
$n+1$-tuples $(q_{1},\dots,q_{n},q)$ of integers , where $q\not=0$ :

$u_{(q_{1},\dots,q_{n})}(x_{1},\dots,x_{n})=q_{1}x_{1}+q_{2}x_{2}+\dots+q_{n}x_{n}\not\equiv
0,$ $v_{(q_{1},\dots,q_{n},q)}(x_{1},\dots,x_{n})=\forall
y(q_{1}x_{1}+q_{2}x_{2}+\dots+q_{n}x_{n}+qy\equiv 0\\\
\Longrightarrow\bigvee_{|k_{i}|\leq|\frac{q_{i}}{q}|,i=1,\dots n}y\equiv
k_{1}x_{1}+\dots+k_{n}x_{n}).$

Every such formula is satisfied in $H$ by the tuple
$\bar{h}=(h_{1},\dots,h_{n})$. Indeed, for the formulas
$u_{(q_{1},\dots,q_{n})}(x_{1},\dots,x_{n})$ this statement is obvious.
Consider the formulas $v_{(q_{1},\dots,q_{n},q)}$. Suppose that for an element
$h\in H$ we have $q_{1}h_{1}+q_{2}h_{2}+\dots+q_{n}h_{n}+qh=0$ for some
integers $(q_{1},\dots,q_{n},q)$ and $q\not=0$. Since $(h_{1},\dots,h_{n})$ is
a base, $h=k_{1}h_{1}+\dots+k_{n}h_{n}$ for some integers
$k_{i},\;i=1,\dots,n$. It obvious that $k_{i}=-\frac{q_{i}}{q}$. Thus all
considered formulas belong to $tp^{H}(\bar{h})$.

Since $H$ and $G$ are isotyped, all formulas $u_{(q_{1},\dots,q_{n})}$ and
$v_{(q_{1},\dots,q_{n},q)}$ belong to $tp^{G}(\bar{g})$ for some $n$-tuple
$\bar{g}=(g_{1},\dots,g_{n})$ in $G$. First of all this means that elements
$g_{1},\dots,g_{n}$ are linearly independent. Let $g$ be an arbitrary element
in $G$. Since rank of $G$ is $n$, the elements $g_{1},\dots,g_{n},g$ are
linearly dependent, that is, $q_{1}g_{1}+\dots+q_{n}g_{n}+qg=0$ for some
integers $(q_{1},\dots,q_{n},q)$, which not all are equal to zero. Taking into
account that the first $n$ elements are linearly independent, we conclude that
$q\not=0$. Since $v_{(q_{1},\dots,q_{n},q)}(g_{1},\dots,g_{n})$ is valid in
$G$, we obtain that

$\bigvee_{|k_{i}|\leq|\frac{q_{i}}{q}|,i=1,\dots
n}g=k_{1}g_{1}+\dots+k_{n}g_{n}.$

This means that $g=k_{1}g_{1}+\dots+k_{n}g_{n}$ for some integers
$k_{1},\dots,k_{n}$.

Consequently $G$ it is generated by $g_{1},\dots,g_{n}$, and therefore $G$ is
isomorphic to $H$. ∎

Conjecture. It seems to be probable that analogous result takes place for
nilpotent groups too.

Remark B. Plotkin writes [8] that Z. Sela has proved a similar fact for free
non-commutative groups (unpublished).

## References

  * [1] C. C. Chang, H. J. Keisler: _Model Theory_ , North-Holland Publishing Company (1973).
  * [2] Chloe Perin and Rizos Sklinos: _Homogeneity in the free group_ , Preprint (2005).ArXiv: math.GR/1003.4095v1
  * [3] A.G. Kurosh: _Theory of Groups_ , ”Nauka” (1967)
  * [4] W. Magnus, A. Karrass, D. Solitar: _Combinatorial Group Theory_ , IP (1966)
  * [5] B. Plotkin: _Seven lectures on the universal algebraic geometry_ , Preprint,(2002), Arxiv:math, GM/0204245, 87pp.
  * [6] B. Plotkin: _Algebraic geometry in First Order Logic_ , Sovremennaja Matematika and Applications 22 (2004), p. 16–62. Journal of Math. Sciences, 137, n.5, (2006), p. 5049– 5097. http:// arxiv.org/ abs/ math GM/0312485.
  * [7] B. Plotkin: _Some results and problems related to universal algebraic geometry,_ International Journal of Algebra and Computation, 17(5/6), (2007), p. 1133–1164.
  * [8] B. Plotkin: _Isotyped algebras._ Arxiv: math.LO/0812.3298v2 (2009). Submitted.
  * [9] B. Plotkin, E. Aladova, E. Plotkin: _Algebraic logic and logically-geometric types in varieties of algebras_ , Preprint (2011). ArXiv:math.LO/1108.0573v1
  * [10] B. Plotkin, G. Zhitomirski: _Automorphisms of categories of free algebras of some varieties_ , J. Algebra, 306, (2006), no. 2, p. 344 -367.
  * [11] B. Plotkin, G. Zhitomirski: _On automorphisms of categories of universal algebras_ , Internat. J. Algebra Comput. 17, (2007), no. 5-6, p. 1115–1132.
  * [12] B. Plotkin, G. Zhitomirski: _Some logical invariants of algebras and logical relations between algebras_ , Algebra and Analysis, 19:5, (2007), p. 214–245, St. Peterburg Math. J., 19:5, (2008), p. 859–879.


# Universal behavior of the Covid-19 tails: Inverse power-law distribution

E. Aydiner<EMAIL_ADDRESS>Theoretical and Computational
Physics Research Laboratory, İstanbul University, 34134 İstanbul, Türkiye
Department of Physics, İstanbul University, 34134 İstanbul, Türkiye  E. Yılmaz
<EMAIL_ADDRESS>Theoretical and Computational Physics Research
Laboratory, İstanbul University, 34134 İstanbul, Türkiye  Institute of
Graduate Studies in Science, Istanbul University, 34134 İstanbul, Türkiye

(2 April 2024, İstanbul)

###### Abstract

Power-law distribution is one of the most important laws known in nature. Such
a special universal behavior is known to occur in very few physical systems.
In this work, we analyzed the mortality distribution of the Covid-19 pandemic
tails for different countries and continents to discuss the possible universal
behavior of the pandemic. Surprisingly, we found that the mortality
distribution of Covid-19 follows inverse power-law decays. These universal
behaviors for the pandemic are reported in the present work for the first
time. Additionally, we showed that mortality tails also decay with time
obeying to the inverse power law.

Covid-19, inverse power-law, universal behavior.

##  I Introduction

Power-law distribution is given in the form

$\displaystyle f(x)\sim x^{\pm\alpha}$ (1)

where $\alpha$ is the exponent which depends on the medium and $x$ is the
continuous or stochastic spatial or time variable. The negative sign for
$\alpha$, Eq.(1) is called the inverse power-law distribution.

As it is known, power-law distribution is a law that has universal
characteristics. Power-law distributions occur in an extraordinarily diverse
range of phenomena. This law has universality and exists in various fields,
from physics to biology, from political sciences to economics. For example;
income distribution [1], population of city [2, 3], the sizes of earthquakes
[4], the frequency of use of words in any human language [5], diameter of moon
craters [6], the number of citations received by papers [7], the number of
hits on web pages [8], intensity of solar flares [9], intensity of wars [10],
the number of “hits” received by web sites [8], the frequency of occurrence of
personal names in most cultures [11], the numbers of species in biological
taxa [12], connection distribution of the airports and internet web sites [13]
and so on.

It should be noted that power laws in physics also occur in many situations
other than the statistical distributions of quantities. For instance, Kepler’s
law [14], Newton [15] law for gravity, Coulomb’s famous law, and the
Stefan–Boltzmann law [16, 17] has a power-law form. Similarly, power-law
behavior can be seen in the second-order phase transition where correlation
functions and other physical quantities near the critical points decay with
power-law exponents. Supercritical state of matter and supercritical fluids
where supercritical exponents of heat capacity and viscosity decay with power-
law [18]. In addition to the examples mentioned above, power-law behavior may
occur for various self-organized critical systems [19] such as forest fires
[20, 21] and avalanche phenomena [19].

In this study, we reported that the inverse power law distribution can be seen
in another phenomenon. As known, Covid-19, which started in 2020, was declared
a pandemic by the World Health Organization [22]. Between 2020 and 2023, it
can be said that it is recorded that approximately one billion people were
infected and more than seven million people lost their lives [23]. We examined
the tail of Covid-19, that is, a short region near the end of the Covid-19
pandemic, and we showed that the distribution function in this region complies
with the inverse power-law. In addition, the time dependence of deaths
occurring in this tail region will be examined in detail.

This work is organized as follows: In Section II, we will analyze the power-
law behavior of the Covid-19 morality data tails for various countries and
continents. In Section III, we will analyze the time dependence of the
mortality data of the Covid-19 tails for the same regions. Finally, we close
with some discussion and conclusion in the last section.

##  II Power-law distribution of the Covid-19 tails

We analyze eight countries and continental data for Covid-19 tails in the
present work. Firstly, we consider Germany (DEU), Espany (ESP),

Figure 1: Mortality distribution $P(m)$ versus mortality frequency $m$ for
eight countries in (a) DEU, in (b) ESP, in (c) FRA, in (d) GBR, in (e) GRC, in
(f) ITA, in (g) NLD, and in (h) TUR.

France (FRA), United Kingdom (GBR), Greece (GRC), Italy (ITA), Holland (NLD),
and Türkiye (TUR) mortality data for the period of the end of the pandemic
[23]. We plotted the data of these countries at $\log-\log$ scale in Fig.1. As
can be seen from the figure, the data of these examined countries comply with
the power law given in Eq.(1):

$\displaystyle P(m)\sim m^{-\alpha}\quad\text{with}\quad 0<\alpha<2$ (2)

where $\alpha$ was obtained separately for each country. This universal
behavior is known as the inverse power-law.

Figure 2: Mortality distribution $P(m)$ versus mortality frequency $m$ for six
continentals in (a) Africa, in (b) Asia, in (c) Europe, in (d) North America,
in (e) Oceania and in (f) South America.

As can be seen from the sub-figures, $\alpha$ values take values between 0 and
2 for countries. For instance; for DEU $\alpha=1.0347$, for ESP
$\alpha=1.0347$, for FRA $\alpha=1.0347$, for GBR $\alpha=1.0347$, for GRC
$\alpha=1.0347$, for ITA $\alpha=1.0347$, for NLD $\alpha=1.0347$, and for TUR
$\alpha=1.0347$.

It is worth noting that the value of the exponents may not be related to the
spread and death rate of Covid-19 for every country. The value of this
parameter is valid only for the Covid-19 queue. We should state that this
parameter represents the distribution of deaths in this tail region according
to death frequency.

We also looked at the death distributions for Covid-19 outbreaks in six
different continents to test whether the power-law distribution obtained for
countries varies for wider geographies, in other words, whether this
characteristic behavior is universal. It can be seen that the distribution
functions of the continents given in Fig.2 also comply with Eq.(2). For
continents, similarly, the $\alpha$ value takes values between 0 and 2. For
instance: for Africa $\alpha=1.1857$, for Asia $\alpha=1.1307$, for Europe
$\alpha=0.88946$, for North America $\alpha=0.91022$, for Oceania
$\alpha=1.14748$, for South America $\alpha=$1.0341.

Our results on the Covid-19 tail show that death distributions during the
pandemic termination period have a universal character and obey the inverse
power law, as in many interesting phenomena in nature [1, 2, 3, 4, 5, 6, 7, 8,
9, 10, 8, 11, 12, 13]. This universal behavior was obtained for the first time
in this study of the Covid-19 tail.

##  III Time-dependent Decay of the Covid-19 tails

As in the previous section, we analyze eight same countries and the same
continental data to discuss the time dependence of the Covid-19 tails.
Therefore, we analyze the time dependence of the Covid-19 data tails for the
same countries given in the previous section.

Figure 3: Time-dependence of the mortality $M(t)$ versus time $t$ for eight
countries in (a) DEU, in (b) ESP, in (c) FRA, in (d) GBR, in (e) GRC, in (f)
ITA, in (g) NLD, and in (h) TUR.

It is important to see how the deaths resulting from the Covid-19 epidemic in
these countries have decreased over time recently. Because we showed that the
death distributions for this period obeyed the inverse power law. Therefore,
the time-dependent decay of this period is a sign that it may comply with a
universal law. Time dependence for these countries is given in Fig.3. These
results show that the time-dependent decay of the Covid-19 tail obeys the
power law:

$\displaystyle M(t)\sim t^{-\beta}$ (3)

where $\beta$ is the decay exponent.

As can be seen from the sub-figures in Fig.3, $\beta$ takes values in the
range $0<\beta<2$ for different countries. For instance: for DEU
$\beta=1.0347$, for ESP $\beta=0.58675$, for FRA $\beta=0.63954$, for GBR
$\beta=0.52065$, for GRC $\beta=0.81123$, for ITA $\beta=0.8899$, for NLD
$\beta=0.8765$, and for TUR $\beta=1.1777$.

Figure 4: Time-dependence of the mortality $M(t)$ versus time $t$ for six
continentals in (a) Africa, in (b) Asia, in (c) Europe, in (d) North America,
in (e) Oceania and in (f) South America.

Similarly, we also analyze the time-dependent behavior of the mortality for
six continents. It can be seen in Fig.4 also complies with Eq.(3). For
continents, the $\beta$ value takes values between 0 and 2. For instance: for
Africa $\beta=1.7003$, for Asia $\beta=1.4827$, for Europe $\beta=1.1856$, for
North America $\beta=0.68167$, for Oceania $\beta=0.56571$, for South America
$\beta=1.2028$.

One can see that the time-dependent decay or relaxation in deaths in the
Covid-19 tails also follows the power law like mortality distributions. As is
known, relaxation processes can be modeled in terms of diffusion. If we
evaluate the problem by associating it with the diffusion process, we can
classify the time-dependent behavior of the Covid-19 tail as follows:

$M(t)\sim t^{-\beta},\quad\begin{cases}\beta=1&\text{normal diffusion}\\\
0\leq\beta<1&\text{sub-diffusion}\\\ 1<\beta\leq 2&\text{super-
diffusion}\end{cases}$ (4)

We reported these novel and interesting results for Covid-19 tails for the
first time in the present study.

##  IV Conclusion

In this study, it is the first time, we analyzed the tails of Covid-19 for
eight countries and six continental regions to discuss the universal behavior
in the distribution and time dependence of the tails. Firstly, we show that
the deaths in the tail of Covid-19 for the different countries and different
regions we analyzed can be represented by a distribution function that obeys
the inverse power law: $P(m)\sim m^{-\alpha}$ with $0<\alpha<2$. Secondly, we
analyzed the time dependence of the tail of Covid-19 death data for the same
countries and the same continents and showed that the number of deaths
decreases with an inverse power law: $M(t)\sim t^{-\beta}$ with $0<\beta<2$.
These novel and interesting results are first reported in this work.

It should be pointed out that the inverse power law is universal, therefore
the obtained results indicate that the tail of Covid-19 has a distribution
function of the mortality and mortality relaxation with time to obey this
universal law. On the other hand, we analyzed the mortality peaks in the
entire time series as they increased and decreased, but we did not observe a
similar universal behavior. The results we obtained are only valid in the
Covid-19 tail, that is, in regions very close to the end of the pandemic.

To the best of our knowledge, power law distribution and decay mechanisms are
a controversial issue. Many alternative mechanisms have been suggested to
generate power-law behavior. For example, it has been shown in various studies
that the sum of different exponential distributions can generate a power-law
distribution [24, 25], and that a random walk and critical phenomena [26] or
taxonomy model of biological species [27, 28] can generate a power-law
behavior. Additionally, it is possible to add different models [29, 30, 31] to
them. We do not currently provide a mathematical model of how various
stochastic effects produce power-laws in the tail of Covid-19 in our example,
however, we can discuss possible predictions. The Covid-19 pandemic, which
started in 2020 and infected approximately one billion people and caused more
than seven million deaths by the end of 2023, has been thoroughly suppressed
thanks to the preventive measures of governments and the developed vaccines.
Despite the emergence of different variants, the virus rose to an acceptable
level of herd immunity and lost its effect. The tail region, where Covid-19
begins to lose its effect, is therefore expected to differ significantly from
the dynamics of the intermediate waves in the time series. Because we can
evaluate that the fact that all stochastic parameters that play a role in
suppressing the lethality of Covid-19 in the tail region have been maximized
may have played a role in generating the power-law dynamics.

As a result, as known that power-law dynamics is an active research topic in
many fields of science such as physics, computer science, linguistics,
geophysics, neuroscience, sociology, economy and more. Therefore, it is very
important efforts to investigate the origins of power-law relations and to
observe and verify them in physics and other areas. In this study, we have
shown that the power-law dynamic can also take place in a pandemic.

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