Schauffler-Like Theorems for Medial and Paramedial Algebras

Dr. D. N. Harutyunyan,

IntelliPaper

Submission received 1 July 2023
Peer Review Double Blind Handling Editor
Accepted 5 July 2023 Access Review Comments
Online published 31 July 2023

− Abstract

In [1], The endolinearity of regular division binary algebras satisfying second-order associativity identities was shown. Furthermore, schaufler- like theorems were proven for these algebras. This paper aims to establish similar results for regular division binary algebras satisfying second-order identities of mediality or paramediality. MSC 2010: 03C05, 03C85, 20N05.

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I. INTRODUCTION

We call the division(cancelation) groupoid if for any the left and right multiplications are surjections(injections), if the groupoid is both division and cancellation then it's called quasi-group. If is a division(cancellation) groupoid, then its operation is called a divisible(cancellable) operation. A binary algebra is called division(cancellation) if each operation is a divisible operation and it's called invertible algebra if it's both division and cancellable algebra. We call a groupoid left-regular if , where . Similarly, we define the right-regular groupoid. We call a groupoid regular if it is simultaneously left-regular and right-regular. If is a regular groupoid, then its operation is called regular. A binary algebra is referred to as regular if each operation is a regular operation. We say that a groupoid is homotopic to a groupoid if there exist such mappings that the equality holds for any [2, 3]. Then the triple is called a homotopy from to . If , then the groupoids are called principally homotopic. If are surjective mappings, then the groupoids are called epitopic or principally epitopic, respectively. We say that an algebra is homotopic (epitopic) to a groupoid if for each the groupoid .

is homotopic (epitopic) to the groupoid . In the same manner we define the principal homotopy (epitopy) of an algebra to a groupoid . An algebra is referred to as r-algebra if it is regular, division, and there exists at least one invertible operation . A binary algebra is called left(right)-linear on a groupoid if each its operation is left (right) linear on the groupoid , that is, for each operation there exists an automorphism of the groupoid and a permutation of the set Q such that:

A (x, y) = \phi x \cdot \alpha y,

(A(x,y) = \alpha x \cdot \phi y)

binary algebra is called linear (endolinear) on a groupoid if each its operation is linear (endolinear) on the groupoid , that is, for each operation there exist automorphisms(endomorphisms) and of the groupoid and an element such that

A(x,y) = (\phi_{A} x \cdot t_{A}) \cdot{}_{A} y

for any .

During World War II, while working at the German cryptographic center, Schauffler obtained applications In Cryptography using invertible algebras that satisfy second-order identities, through proving the following theorem. [4-6]

Theorem 1.1 (Schauffler). Let Q be a non-empty set. The following propositions are equivalent:

For all , quasigroups, there exist , quasigroups, such that following -identity holds:

\forall X, Y \exists X ^ {\prime}, Y ^ {\prime} \forall x, y, z (X (Y (x, y), z) = X ^ {\prime} (x, Y ^ {\prime} (y, z))),\tag{1.1}

For all , quasigroups, there exist , quasigroups, such that following -identity holds:

\forall X, Y \exists X ^ {\prime}, Y ^ {\prime} \forall x, y, z (X (x, Y (y, z)) = X ^ {\prime} (Y ^ {\prime} (x, y), z)),\tag{1.2}

.

In the [7] proved schauffler-like theorem for other second-order identities and hyperintensities (see [8-10]).

Theorem 1.2. (Movsisyan) Let Q be non empty set. The following propositions are equivalent:

for all , quasigroups, there exist , quasigroups, such that (1.1) identity holds,
for all , quasigroups, there exist , quasigroups, such that (1.2) identity holds,
for all , loops, there exist , quasi-groups, such that (1.1) identity holds,
for all , loops, there exist , quasi-groups, such that (1.2) identity holds,
for all , loops, there exist , loops, such that (1.1) identity holds,
for all , loops, there exist , loops, such that (1.2) identity holds,
following hyperidentity holds in the algebra:

\[X (x, Y (y, z)) = X (Y (x, y), z),\]

following hyperidentity holds in the algebra:

\[X (x, Y (y, z)) = X (Y (x, y), z),\]

For all quasigroup, there exist , quasi-groups, such that following -identity holds:

\forall X \exists X ^ {\prime}, Y ^ {\prime} \forall x, y, z (X (X (x, y), z) = X ^ {\prime} (x, Y ^ {\prime} (y, z))),\tag{1.3}

For all quasigroup, there exist , quasi-groups, such that following -identity holds:

\forall X \exists X ^ {\prime}, Y ^ {\prime} \forall x, y, z (X (x, X (y, z)) = X ^ {\prime} (Y ^ {\prime} (x, y), z)),\tag{1.4}

where is the set of all loop-operations over the set Q.

In [11] it was proved that an invertible algebra with the formula (1.1) or (1.2) is linear on the group. In this paper, we will prove that r-algebras with second-order formulas of mediality and paramediality are endolinear on the group.

II. PRELIMANRY RESULTS

Definition 1. A mapping is called quasiendomorphism of a group if for all , where 1 is the unity of the group . If is also a bijection from to , then is called the quasi-automorphism of the group .

Lemma 2.1. Each quasiendomorphism of a group has the form , where , , and is an endomorphism of the group . The converse is valid: if is an endomorphism of a group , then an arbitrary mapping from to is a quasiendomorphism of the group .

Proof. Suppose that . We show that is an endomorphism. We have

\phi^ {\prime} (a b) = L _ {k ^ {- 1}} \phi (a b) = k ^ {- 1} \cdot \phi a \cdot (\phi 1) ^ {- 1} \cdot \phi b = (k ^ {- 1} \cdot \phi a) \cdot (k ^ {- 1} \cdot \phi b) = \phi^ {\prime} a \cdot \phi^ {\prime} b.

Lemma 2.2. Suppose that is a group and is the principal epitopy of this group. Then is a surjective quasiendomorphism of this group; moreover, if

\alpha (x \cdot y) = \beta x \cdot \gamma y,\tag{2.5}

then and are also quasiendomorphisms.

Proof. Making the successive replacements in the (2.5): (1) , (2) , and (3) , we obtain

\alpha y = \beta 1 \cdot \gamma y, x = \beta x \cdot \gamma 1, \alpha 1 = \beta 1 \cdot \gamma 1.\tag{2.6}

We transform equality (2.5), taking into account equalities (2.6):

\alpha (x \cdot y) = \alpha x \cdot (\gamma 1) ^ {- 1} \cdot (\beta 1) ^ {- 1} \cdot \alpha y = \alpha x \cdot (\beta 1 \cdot \gamma 1) ^ {- 1} \cdot \alpha y = \alpha x \cdot (\alpha 1) ^ {- 1} \cdot \alpha y,

that is, is a quasiendomorphism.

From (2.5) we also have

\begin{array}{c} \beta (x \cdot y) = \alpha (x \cdot y) \cdot (\gamma 1) ^ {- 1} = \alpha x \cdot (\alpha 1) ^ {- 1} \cdot (\alpha y \cdot (\gamma 1) ^ {- 1}) = \\(\alpha x \cdot (\gamma 1) ^ {- 1}) \cdot (\beta 1) ^ {- 1} \cdot \beta y = \beta x \cdot (\beta 1) ^ {- 1} \cdot \beta y. \end{array}

In the same manner, we prove that is a quasiendomorphism of the group .

Lemma 2.3. [12] If a loop is principally homotopic to a group , then they are isomorphic. If a group is principally homotopic to a group , then they are isomorphic.

Theorem 2.3. [12] Let the set Q form a division groupoid under the six operations (for ) and or is regular operation. If these operations satisfy the following equation:

A _ {1} (A _ {2} (x, y), A _ {3} (u, v)) = A _ {4} (A _ {5} (x, u), A _ {6} (y, v)),\tag{2.7}

for all elements , then there exists an operation ( ) under which Q forms an abelian group and all these six division groupoids are epitopic to the group and there exist eight surjective mappings of Q onto itself such that:

A _ {1} (x, y) = \alpha x \cdot \phi y,

\alpha A _ {2} (x, y) = \gamma x \cdot \delta y,

\phi A _ {3} (x, y) = \lambda x \cdot \beta y,

A _ {4} (x, y) = \psi x \cdot \sigma y,

\psi A _ {5} (x, y) = \gamma x \cdot \lambda y,

\sigma A _ {6} (x, y) = \delta x \cdot \beta y.

The abelian group is unique up to isomorphisms.

Theorem 2.4. [12] Let the set Q form a division groupoid under the six operations (for ) and or is regular operation. If these operations satisfy the following paramedial equation:

A _ {1} (A _ {2} (x, y), A _ {3} (u, v)) = A _ {4} (A _ {5} (v, y), A _ {6} (u, x))\tag{2.8}

for all elements , then there exists an operation ( ) under which Q forms an abelian group, and all these six division groupoids are epitopic to the group and there exist eight surjective mappings of Q onto itself such that:

A _ {1} (x, y) = \alpha x \cdot \phi y,

\alpha A _ {2} (x, y) = \gamma x \cdot \delta y,

\phi A _ {3} (x, y) = \lambda x \cdot \beta y,

A _ {4} (x, y) = \sigma x \cdot \psi y,

\sigma A _ {5} (x, y) = \beta x \cdot \delta y,

\psi A _ {6} (x, y) = \lambda x \cdot \gamma y.

The abelian group is unique up to isomorphisms.

III. ENDO LINEAR REPRESENTATIONS

Theorem 3.5. Suppose that is an r-algebra. If for arbitrary there exist such that the following identity of mediality holds:

X (Y (x, y), Y (u, v)) = X ^ {\prime} (Y ^ {\prime} (x, u), Z ^ {\prime} (y, v)),\tag{3.9}

then there exist an abelian group such that an arbitrary operation is endolinear over the group . The group is determined uniquely up to isomorphism.

Proof. Let be invertable operation, then from the theorem 2.3 we will have that exists operations and abelian group such that following identities hold:

\begin{array}{c} X (x, y) = \alpha x \cdot \phi y, \\\alpha X (x, y) = \gamma x \cdot \delta y, \\\phi X (x, y) = \lambda x \cdot \beta y, \\X _ {1} (x, y) = \psi x \cdot \sigma y, \\\psi X _ {2} (x, y) = \gamma x \cdot \lambda y, \\\sigma X _ {3} (x, y) = \delta x \cdot \beta y. \end{array}

Since X is invertible operation, then we will have that and are bijections, moreover:

\alpha (\alpha x \cdot \phi y) = \gamma x \cdot \delta y,

from which we will obtain:

\alpha (x \cdot y) = \gamma \alpha^ {- 1} x \cdot \delta \phi^ {- 1} y,

This means that is quasiautomorphism of the group .

Lets fix operation X, for every operation Y exists operations such that (3.9) identity holds, and then from the theorem 2.3 we will have that exist abelian group such that those identities hold:

X(x,y)=\alpha_{Y}x\cdot_{Y}\phi_{Y}y,

\alpha_{Y}Y(x,y)=\gamma_{Y}x\cdot_{Y}\delta_{Y}y,

\phi_{Y}Y(x,y)=\lambda_{Y}x\cdot_{Y}\beta_{Y}y,

X^\prime(x,y)=\gamma_{Y}x\cdot_{Y}\sigma_{Y}y,

Y^\prime(x,y)=\gamma_{Y}x\cdot_{Y}\lambda_{Y}y,

\sigma_ {Y} Z ^ {\prime} (x, y) = \delta_ {Y} x \cdot_ {Y} \beta_ {Y} y.

From the proof of the theorem 2.3 we can construct the group in such way that .

We have that: , which is the same as: , where is right inverse of .

We have for every operation the following identity is true:

\begin{array}{c} \alpha Y (x, y) = \gamma_ {Y} x \cdot_ {Y} \delta_ {Y} y = \gamma_ {Y} x \cdot \delta_ {Y} \phi h _ {\phi_ {Y}} y \implies \\Y (x, y) = \alpha^ {- 1} (\gamma_ {Y} x \cdot \delta_ {Y} \phi h _ {\phi_ {Y}} y). \end{array}

Since the set of all quasiautomorphisms of the group is also a group, then will also be quasiautomorphisms, which means:

Y(x,y) = \alpha^{-1}\gamma_Y x\cdot(\alpha^{-1}e)^{-1}\cdot\alpha^{-1}\delta_Y\phi h_{\phi_Y} y = \alpha^{-1}\gamma_Y x\cdot L_{(\alpha^{-1}e)^{-1}}\alpha^{-1}\delta_Y\phi h_{\phi_Y} y,

where is the identity element of the group , is left translation of the group with the element.

We obtained that for every operation there exists surjections and , such that . This means we can rewrite the representations of the operations in the following way.

\left\{ \begin{array}{l} X (x, y) = \alpha_{X} x \cdot \beta_{X} y, \\ Y (x, y) = \alpha_{Y} x \cdot \beta_{Y} y, \\ X^{\prime} (x, y) = \alpha_{X^{\prime}} x \cdot \beta_{X^{\prime}} y, \\ Y^{\prime} (x, y) = \alpha_{Y^{\prime}} x \cdot \beta_{Y^{\prime}} y, \\ Z^{\prime} (x, y) = \alpha_{Z^{\prime}} x \cdot \beta_{Z^{\prime}} y \end{array} \right.

where are surjections.

By doing the replacements in the identity (3.9) we will obtain:

\alpha_ {X} \left(\alpha_ {Y} x \cdot \beta_ {Y} y\right) \cdot \beta_ {X} \left(\alpha_ {Y} u \cdot \beta_ {Y} v\right) = \alpha_ {X ^ {\prime}} \left(\alpha_ {Y ^ {\prime}} x \cdot \beta_ {Y ^ {\prime}} u\right) \cdot \beta_ {X ^ {\prime}} \left(\alpha_ {Z ^ {\prime}} y \cdot \beta_ {Z ^ {\prime}} v\right).

Replacing , , and , where respectively are the right inverses of the and e is the identity element of the group , we will have:

\beta_ {X} (u \cdot v) = \alpha_ {X ^ {\prime}} (\alpha_ {Y ^ {\prime}} h _ {\alpha_ {Y}} h _ {\alpha_ {X}} e \cdot \beta_ {Y ^ {\prime}} h _ {\alpha_ {Y}} u) \cdot \beta_ {X ^ {\prime}} (\alpha_ {Z ^ {\prime}} h _ {\beta_ {Y}} e \cdot \beta_ {Z ^ {\prime}} h _ {\beta_ {Y}} v) = \mu u \cdot \nu v,

where and . We showed that is quasiendomorphism of the group and from lemma 2.1 we know that there exists endomorphism of the group such that , where is left translation of the group with the element . We will have following representation of the arbitrary operation where .

By doing following replacements in the identity , , and , we will obtain that is also a quasiendomorphism of the group and from lemma 2.1 we know that there exists endomorphism of the group such that , where is right translation of the group with the element , so we will have:

X(x,y) = \phi_{X} x \cdot b \cdot _{X} y

for every .

Corollary 1. Suppose that is an r-algebra. If for arbitrary there exist such that the following identity holds:

X (Y (x, y), Z (u, v)) = X ^ {\prime} \left(Y ^ {\prime} (x, u), Z ^ {\prime} (y, v)\right),\tag{3.10}

then there exist an abelian group such that an arbitrary operation is endolinear over the group . The group is determined uniquely up to isomorphism.

Corollary 2. Suppose that is an r-algebra. If for arbitrary there exist such that the following identity holds:

X (Y (x, y), Y (u, v)) = X ^ {\prime} \left(Y ^ {\prime} (x, u), Y ^ {\prime} (y, v)\right),\tag{3.11}

then there exist an abelian group such that an arbitrary operation is endolinear over the group . The group is determined uniquely up to isomorphism.

Similarly, we can prove the following results.

Theorem 3.6. Suppose that is an r-algebra. If for arbitrary there exist such that the following identity of paramediality holds:

X (Y (x, y), Y (u, v)) = X ^ {\prime} (Y ^ {\prime} (v, y), Z ^ {\prime} (u, x)),\tag{3.12}

then there exists an abelian group such that an arbitrary operation is endolinear over the group . The group is determined uniquely up to isomorphism.

Corollary 3. Suppose that is an r-algebra. If for arbitrary there exist such that the following identity holds:

X (Y (x, y), Z (u, v)) = X ^ {\prime} \left(Y ^ {\prime} (v, y), Z ^ {\prime} (u, x)\right),\tag{3.13}

then there exists an abelian group such that an arbitrary operation is endolinear over the group . The group is determined uniquely up to isomorphism.

Corollary 4. Suppose that is an r-algebra. If for arbitrary there exist such that the following identity holds:

X (Y (x, y), Y (u, v)) = X ^ {\prime} \left(Y ^ {\prime} (v, y), Y ^ {\prime} (u, x)\right),\tag{3.14}

then there exists an abelian group such that an arbitrary operation is endolinear over the group . The group is determined uniquely up to isomorphism.

Theorem 3.7. Suppose that is an r-algebra. If for arbitrary there exist such that (3.10) identity of mediality satisfies, then there exists an abelian group such that an arbitrary operation is endolinear over the group . The group is determined uniquely up to isomorphism.

Proof. Let's fix , from the theorem 2.3 we will have that there exists operations and abelian group such that following identities hold:

X (x, y) = \alpha x \cdot \phi y,

\alpha Y _ {1} (x, y) = \gamma x \cdot \delta y,

\phi Z _ {1} (x, y) = \lambda x \cdot \beta y,

X (x, y) = \psi x \cdot \sigma y,

\psi Y _ {2} (x, y) = \gamma x \cdot \lambda y,

\sigma Z _ {2} (x, y) = \delta x \cdot \beta y.

Lets fix operation X, and for every opreation we have that there exists opreations such that identity (3.10) holds, and from the theorem 2.3 we will have that there exists surjections and abelian group such that following identities hold:

X (x, y) = \alpha_ {X ^ {\prime}} x \cdot_ {X ^ {\prime}} \phi_ {X ^ {\prime}} y,

\alpha_{X^{'}} Y (x, y) = \gamma_{X^{'}} x \cdot_{X^{'}} \delta_{X^{'}} y,

\phi_{X^{'}} Z(x, y) = \lambda_{X^{'}} x \cdot_{X^{'}} \beta_{X^{'}} y,

X^\prime(x,y) = \mathbf{\nabla}_{X^\prime} x \cdot_{X^\prime} \sigma_{X^\prime} y,

_{X^{ ext{'}}}Y^{ ext{'}}(x,y) = \gamma_{X^{ ext{'}}} x \cdot_{X^{ ext{'}}} \lambda_{X^{ ext{'}}} y,

\sigma_ {X ^ {\prime}} Z ^ {\prime} (x, y) = \delta_ {X ^ {\prime}} x \cdot_ {X ^ {\prime}} \beta_ {X ^ {\prime}} y.

By doing following replacements and we will obtain:

x \cdot_ {X ^ {\prime}} y = \alpha h _ {\alpha_ {X ^ {\prime}}} x \cdot \phi h _ {\phi_ {X ^ {\prime}}} y,

where and are surjections.

We showed that arbitrary operations has following representation:

X ^ {\prime} (x, y) = \quad_ {X ^ {\prime}} x \cdot_ {X ^ {\prime}} \sigma_ {X ^ {\prime}} y = \alpha h _ {\alpha_ {X ^ {\prime}}} \quad_ {X ^ {\prime}} x \cdot \phi h _ {\phi_ {X ^ {\prime}}} \sigma_ {X ^ {\prime}} y = \nu_ {X ^ {\prime}} x \cdot \mu_ {X ^ {\prime}} y,

where and are surjections.

From which we have that there exists abelian group that following identieis hold:

\left\{ \begin{array}{l} X (x, y) = \nu_{X} x \cdot \mu_{X} y, \\ Y (x, y) = \nu_{Y} Y x \cdot \mu_{Y} y, \\ Z (x, y) = \nu_{Z} x \cdot \mu_{Z} y, \\ X^{\prime} (x, y) = \nu_{X^{\prime}} x \cdot \mu_{X^{\prime}} y, \\ Y^{\prime} (x, y) = \nu_{Y^{\prime}} x \cdot \mu_{Y^{\prime}} y, \\ Z^{\prime} (x, y) = \nu_{Z^{\prime}} x \cdot \mu_{Z^{\prime}} y \end{array} \right.

where are surjections.

By doing replacements in the identity we will have:

\nu_ {X} (\nu_ {Y} x \cdot \mu_ {Y} y) \cdot \mu_ {X} (\nu_ {Z} u \cdot \mu_ {Z} v) = \nu_ {X ^ {\prime}} (\nu_ {Y ^ {\prime}} x \cdot \mu_ {Y ^ {\prime}} u) \cdot \mu_ {X ^ {\prime}} (\nu_ {Z ^ {\prime}} y \cdot \mu_ {Z ^ {\prime}} v).

Replacing , , and , where respectively are the right inverses of the and e is the identity element of the group , we will have:

, where and . We showed that is quasiendomorphism of the group and from lemma 2.1 we know that there exists endomorphism of the group such that , where is left translation of the group with the element . We will have following representation of the arbitrary operation : where .

By doing following replacements in the identity , , and , we will obtain that is also a quasiendomorphism of the group and from lemma 2.1 we know that there exists endomorphism of the group such that , where is right translation of the group with the element , so we will have:

X (x, y) = \phi_ {X} x \cdot b \cdot \quad_ {X} y,

Theorem 3.8. Suppose that is an r-algebra. If for arbitrary there exist such that (3.13) identity of paramediality satisfies, then there exists an abelian group such that an arbitrary operation is endolinear over the group . The group is determined uniquely up to isomorphism.

Let denote by all the regular division operations of the set .

Theorem 3.9. One of the following -identities of mediality

\forall X, Y \exists X ^ {\prime}, Y ^ {\prime}, Z ^ {\prime} \forall x, y, u, v X (Y (x, y), Y (u, v)) = X ^ {\prime} (Y ^ {\prime} (x, u), Z ^ {\prime} (y, v)),

\forall X, Y, Z \exists X ^ {\prime}, Y ^ {\prime}, Z ^ {\prime} \forall x, y, u, v X (Y (x, y), Z (u, v)) = X ^ {\prime} \left(Y ^ {\prime} (x, u), Z ^ {\prime} (y, v)\right),

\forall X, Y \exists X ^ {\prime}, Y ^ {\prime} \forall x, y, u, v X (Y (x, y), Y (u, v)) = X ^ {\prime} (Y ^ {\prime} (x, u), Y ^ {\prime} (y, v)),

\forall X, X ^ {\prime} \exists Y, Y ^ {\prime}, Z, Z ^ {\prime} \forall x, y, u, v X (Y ^ {\prime} (x, y), Y ^ {\prime} (u, v)) = Y (X ^ {\prime} (x, u), X ^ {\prime} (y, v)),

or paramediality

\forall X, Y \exists X ^ {\prime}, Y ^ {\prime}, Z ^ {\prime} \forall x, y, u, v X (Y (x, y), Y (u, v)) = X ^ {\prime} (Y ^ {\prime} (v, y), Z ^ {\prime} (u, x)),

\forall X, Y, Z \exists X ^ {\prime}, Y ^ {\prime}, Z ^ {\prime} \forall x, y, u, v X (Y (x, y), Z (u, v)) = X ^ {\prime} (Y ^ {\prime} (v, y), Z ^ {\prime} (u, x)),

\forall X, Y \exists X ^ {\prime}, Y ^ {\prime} \forall x, y, u, v X (Y (x, y), Y (u, v)) = X ^ {\prime} (Y ^ {\prime} (v, y), Y ^ {\prime} (u, x)),

\forall X, X ^ {\prime} \exists Y, Y ^ {\prime}, Z, Z ^ {\prime} \forall x, y, u, v X (Y (x, y), Z (u, v)) = X ^ {\prime} (Y ^ {\prime} (v, y), Z ^ {\prime} (u, x)),

holds in the algebra , if and only if .

Proof. Let us prove it for the first identity; the rest are proved similarly. It follows from Theorem 3.6 that there exists a group such that any operation is endolinear over this group, which implies that all loops in are endolinear over this group; therefore, they are principally homotopic to this group. From Lemma 2.1 it follows that they are isomorphic to this group; however, the Albert theorems () imply that a nonassociative loop is not isomorphic to a group; therefore, . It is well-known that on a finite set each surjection is a bijection; so, each regular division operation is an invertible operation, that is, any operation in is invertible this means if we fix in the identity we will have:

X (Y (x, y), \alpha v) = X ^ {\prime} (\beta x, Y ^ {\prime} (y, v)),

which is same

X (Y (x, y), v) = X ^ {\prime} (\beta x, Y ^ {\prime} (y, \alpha^ {- 1} v)),

where are bijections ad is inverse of the .

This means satisfies to the following second-order identity of associativity:

\forall X, Y \exists X ^ {\prime \prime}, Y ^ {\prime \prime} \forall x, y, z X (Y (x, y), z) = X ^ {\prime \prime} (x, Y ^ {\prime \prime} (y, z)),

and from the Theorem 1.1 we have that .

The sufficiency follows from the [7].

Funding

The first author was partially supported by the Science Committee of Funding of the Republic of Armenia, project nos. 10-3/1-41 and 21T-1A213.

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The authors declare no conflict of interest.

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Not applicable

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The datasets used in this study are openly available at [repository link] and the source code is available on GitHub at [GitHub link].

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This work did not receive any external funding.

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