# A Complete classification of two-dimensional endo-commutative algebras over
an arbitrary field

D.Asrorov, U.Bekbaev, I.Rakhimov  1Samarqand State University, Samarqand,
Uzbekistan
2Turin Polytechnic University in Tashkent, Tashkent, Uzbekistan
3Universiti Teknologi MARA (UiTM), Shah Alam, Malaysia

###### Abstract.

In the paper, we consider the class of so-called endo-commutative algebras.
From the identity imposed to specify this class, one can easily see that the
product in this class preserves the square of elements. We give a complete
classification of this class, up to isomorphism, over any basic field in
dimension two. This elementary and self-contained exposition uses the recent
result of one of the authors on description of the category of all
$2$-dimensional algebras over any basic field.

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Key words: endo-commutative algebra, isomorphism, structure constants

##  1\. Introduction

The classification problem, up to isomorphism, of a given class of algebras is
one of the important and difficult problems of algebra. So far two approaches
are known to the solution of the problem. One of them is structural (basis
free, invariant) approach. For instance, the classification of finite
dimensional simple and semi-simple associative algebras by Wedderburn and
simple and semi-simple Lie algebras by Cartan are examples of such approach.
But it is observed that this approach becomes more difficult when one
considers more general types of algebras. Another approach to the solution of
the problem is coordinate based. Many researchers have used this type of
approach to classify various, mainly finite dimensional, classes of algebras:
associative [17, 18], Lie [12, 19, 20, 21, 22], Jordan [13] and Leibniz [4, 8,
10, 15]. These two approaches are somehow complementary to each other.

There were attempts to classify all fixed-dimensional algebras, for example,
in [23] a classification of all $2$-dimensional algebras, by basis free
approach, was stated over any basic field. One disadvantage of the basis free
approach is that an application of the obtained classification result to
classification of a given class of algebras is hardly possible. In this
respect the coordinate based classification has advantage over it. For the
coordinated based approach in classification all $2$-dimensional algebras,
over fields with some constraints, one can see [3, 11, 14] and in [1, 2] its
different applications.

Recently, in [5] a complete classification of all $2$-dimensional algebras
over any basic field has been announced and we have come across with the paper
[26], where the authors gave a complete classification of $2$-dimensional
endo-commutative algebras over the field $\mathbb{Z}_{2}$. We noticed that it
can be done over an arbitrary field by more easy and faster way relying on the
results of [5]. This is the subject of the present paper.

##  2\. Preliminaries

We begin by shortly recalling some concepts which will be using in the paper.

Let $\mathbf{A}$ be an $n$-dimensional algebra over a field $\mathbb{F}$ and
$\mathbf{e}=\\{\mathrm{e}_{1},\mathrm{e}_{2},...,\mathrm{e}_{n}\\}$ be a basis
of the underlying vector space of $\mathbf{A}$. Then on the basis $\mathbf{e}$
the algebra $\mathbf{A}$ is represented by a $n\times n^{2}$ matrix (called
the matrix of structure constant, shortly MSC)

$A=\left(\begin{array}[]{ccccccccccccc}a_{11}^{1}&a_{12}^{1}&...&a_{1n}^{1}&a_{21}^{1}&a_{22}^{1}&...&a_{2n}^{1}&...&a_{n1}^{1}&a_{n2}^{1}&...&a_{nn}^{1}\\\
a_{11}^{2}&a_{12}^{2}&...&a_{1n}^{2}&a_{21}^{2}&a_{22}^{2}&...&a_{2n}^{2}&...&a_{n1}^{2}&a_{n2}^{2}&...&a_{nn}^{2}\\\
...&...&...&...&...&...&...&...&...&...&...&...&...\\\
a_{11}^{n}&a_{12}^{n}&...&a_{1n}^{n}&a_{21}^{n}&a_{22}^{n}&...&a_{2n}^{n}&...&a_{n1}^{n}&a_{n2}^{n}&...&a_{nn}^{n}\end{array}\right)$

as follows

$\mathrm{e}_{i}\cdot\mathrm{e}_{j}=\sum\limits_{k=1}^{n}a_{ij}^{k}\mathrm{e}_{k},\
\mbox{where}\ i,j=1,2,...,n.$

Therefore, the product on $\mathbf{A}$ with respect to the basis $\mathbf{e}$
is written as follows

(2.1) $\mathrm{x}\cdot\mathrm{y}=\mathbf{e}A(x\otimes y)$

for any $\mathrm{x}=\mathbf{e}x,\mathrm{y}=\mathbf{e}y,$ where
$x=(x_{1},x_{2},...,x_{n})^{T},$ and $y=(y_{1},y_{2},...,y_{n})^{T}$ are
column coordinate vectors of $\mathrm{x}$ and $\mathrm{y},$ respectively,
$x\otimes y$ is the tensor(Kronecker) product of the vectors $x$ and $y$. Now
and onward for the product “$\mathrm{x}\cdot\mathrm{y}$” on $\mathbf{A}$ we
use the juxtaposition “$\mathrm{x}\mathrm{y}$”.

If $\mathbf{A}$ is a two-dimensional algebra over a field $\mathbb{F}$ and
$\mathbf{e}=\\{\mathrm{e}_{1},\mathrm{e}_{2}\\}$ is a basis then MSC of
$\mathbf{A}$ looks like

$A=\left(\begin{array}[]{cccc}\alpha_{1}&\alpha_{2}&\alpha_{3}&\alpha_{4}\\\
\beta_{1}&\beta_{2}&\beta_{3}&\beta_{4}\end{array}\right).$

In [5] the following classification, up to isomorphism, of two-dimensional
algebras, over any basic field $\mathbb{F}$, in terms of their MSC, was given.

######  Theorem 1.

Any non-trivial $2$-dimensional algebra over a field $\mathbb{F}$
$(Char(\mathbb{F})\neq 2,3)$ is isomorphic to only one of the following
listed, by their matrices of structure constants, such algebras:

  * •

$A_{1}(\mathrm{c})=\begin{pmatrix}\alpha_{1}&\alpha_{2}&1+\alpha_{2}&\alpha_{4}\\\
\beta_{1}&-\alpha_{1}&1-\alpha_{1}&-\alpha_{2}\end{pmatrix},$ where
$\mathrm{c}=(\alpha_{1},\alpha_{2},\alpha_{4},\beta_{1})\in\mathbb{F}^{4}$

  * •

$A_{2}(\mathrm{c})=\begin{pmatrix}\alpha_{1}&0&0&\alpha_{4}\\\
1&\beta_{2}&1-\alpha_{1}&0\end{pmatrix},$ where
$\mathrm{c}=(\alpha_{1},\alpha_{4},\beta_{2})\in\mathbb{F}^{3}$ and
$\alpha_{4}\neq 0$

  * •

$A_{3}(\mathrm{c})=\begin{pmatrix}\alpha_{1}&0&0&\alpha_{4}\\\
0&\beta_{2}&1-\alpha_{1}&0\end{pmatrix}\simeq\begin{pmatrix}\alpha_{1}&0&0&a^{2}\alpha_{4}\\\
0&\beta_{2}&1-\alpha_{1}&0\end{pmatrix},$ where
$\mathrm{c}=(\alpha_{1},\alpha_{4},\beta_{2})\in\mathbb{F}^{3},$
$a\in\mathbb{F}$ and $a\neq 0$

  * •

$A_{4}(\mathrm{c})=\begin{pmatrix}0&1&1&0\\\
\beta_{1}&\beta_{2}&1&-1\end{pmatrix},$ where
$\mathrm{c}=(\beta_{1},\beta_{2})\in\mathbb{F}^{2}$

  * •

$A_{5}(\mathrm{c})=\begin{pmatrix}\alpha_{1}&0&0&0\\\
1&2\alpha_{1}-1&1-\alpha_{1}&0\end{pmatrix},$
where $\mathrm{c}=\alpha_{1}\in\mathbb{F}$

  * •

$A_{6}(\mathrm{c})=\begin{pmatrix}\alpha_{1}&0&0&\alpha_{4}\\\
1&1-\alpha_{1}&-\alpha_{1}&0\end{pmatrix},$ where
$\mathrm{c}=(\alpha_{1},\alpha_{4})\in\mathbb{F}^{2}$ and $\alpha_{4}\neq 0$

  * •

$A_{7}(\mathrm{c})=\begin{pmatrix}\alpha_{1}&0&0&\alpha_{4}\\\
0&1-\alpha_{1}&-\alpha_{1}&0\end{pmatrix}\simeq\begin{pmatrix}\alpha_{1}&0&0&a^{2}\alpha_{4}\\\
0&1-\alpha_{1}&-\alpha_{1}&0\end{pmatrix},$ where
$\mathrm{c}=(\alpha_{1},\alpha_{4})\in\mathbb{F}^{2},$ $a\in\mathbb{F}$ and
$a\neq 0$

  * •

$A_{8}(\mathrm{c})=\begin{pmatrix}0&1&1&0\\\ \beta_{1}&1&0&-1\end{pmatrix},$
where $\mathrm{c}=\beta_{1}\in\mathbb{F}$

  * •

$A_{9}=\begin{pmatrix}\frac{1}{3}&0&0&0\\\
1&\frac{2}{3}&-\frac{1}{3}&0\end{pmatrix},$

  * •

$A_{10}(\mathrm{c})=\begin{pmatrix}0&1&1&1\\\
\beta_{1}&0&0&-1\end{pmatrix}\simeq\begin{pmatrix}0&1&1&1\\\
\beta_{1}^{{}^{\prime}}(a)&0&0&-1\end{pmatrix},$
where $\mathrm{c}=\beta_{1}\in\mathbb{F},$ the polynomial
$(\beta_{1}t^{3}-3t-1)(\beta_{1}t^{2}+\beta_{1}t+1)(\beta_{1}^{2}t^{3}+6\beta_{1}t^{2}+3\beta_{1}t+\beta_{1}-2)$
has no root in $\mathbb{F}$, $a\in\mathbb{F}$ and
$\beta_{1}^{{}^{\prime}}(t)=\frac{(\beta_{1}^{2}t^{3}+6\beta_{1}t^{2}+3\beta_{1}t+\beta_{1}-2)^{2}}{(\beta_{1}t^{2}+\beta_{1}t+1)^{3}}$

  * •

$A_{11}(\mathrm{c})=\begin{pmatrix}0&0&0&1\\\ \beta_{1}&0&0&0\\\
\end{pmatrix}\simeq\begin{pmatrix}0&0&0&1\\\ a^{3}\beta_{1}^{\pm 1}&0&0&0\\\
\end{pmatrix},$ where the polynomial $\beta_{1}-t^{3}$ has no root in
$\mathbb{F},$ $a,\mathrm{c}=\beta_{1}\in\mathbb{F}$ and $a,\beta_{1}\neq 0$

  * •

$A_{12}(\mathrm{c})=\begin{pmatrix}0&1&1&0\\\
\beta_{1}&0&0&-1\end{pmatrix}\simeq\begin{pmatrix}0&1&1&0\\\
a^{2}\beta_{1}&0&0&-1\end{pmatrix},$ where
$a,\mathrm{c}=\beta_{1}\in\mathbb{F}$ and $a\neq 0$

  * •

$A_{13}=\begin{pmatrix}0&0&0&0\\\ 1&0&0&0\end{pmatrix}.$

######  Theorem 2.

Any non-trivial $2$-dimensional algebra over a field $\mathbb{F}$
$(Char(\mathbb{F})=2)$ is isomorphic to only one of the following listed by
their matrices of structure constants, such algebras:

  * •

$A_{1,2}(\mathrm{c})=\begin{pmatrix}\alpha_{1}&\alpha_{2}&1+\alpha_{2}&\alpha_{4}\\\
\beta_{1}&\alpha_{1}&1+\alpha_{1}&\alpha_{2}\end{pmatrix},$ where
$\mathrm{c}=(\alpha_{1},\alpha_{2},\alpha_{4},\beta_{1})\in\mathbb{F}^{4}$

  * •

$A_{2,2}(\mathrm{c})=\begin{pmatrix}\alpha_{1}&0&0&\alpha_{4}\\\
1&\beta_{2}&1+\alpha_{1}&0\end{pmatrix},$ where
$\mathrm{c}=(\alpha_{1},\alpha_{4},\beta_{2})\in\mathbb{F}^{3}$ and
$\alpha_{4}\neq 0$

  * •

$A_{2,2}(\alpha_{1},0,1)=\begin{pmatrix}\alpha_{1}&0&0&0\\\
1&1&1+\alpha_{1}&0\end{pmatrix},$ where $\alpha_{1}\in\mathbb{F}$

  * •

$A_{3,2}(\mathrm{c})=\begin{pmatrix}\alpha_{1}&0&0&\alpha_{4}\\\
0&\beta_{2}&1+\alpha_{1}&0\end{pmatrix}\simeq\begin{pmatrix}\alpha_{1}&0&0&a^{2}\alpha_{4}\\\
0&\beta_{2}&1+\alpha_{1}&0\end{pmatrix},$ where
$\mathrm{c}=(\alpha_{1},\alpha_{4},\beta_{2})\in\mathbb{F}^{3},$
$a\in\mathbb{F}$ and $a\neq 0$

  * •

$A_{4,2}(\mathrm{c})=\begin{pmatrix}\alpha_{1}&1&1&0\\\
\beta_{1}&\beta_{2}&1+\alpha_{1}&1\end{pmatrix}\simeq\begin{pmatrix}\alpha_{1}&1&1&0\\\
\beta_{1}+(1+\beta_{2})a+a^{2}&\beta_{2}&1+\alpha_{1}&1\end{pmatrix},$
where $\mathrm{c}=(\alpha_{1},\beta_{1},\beta_{2})\in\mathbb{F}^{3}$

  * •

$A_{5,2}(\mathrm{c})=\begin{pmatrix}\alpha_{1}&0&0&\alpha_{4}\\\
1&1+\alpha_{1}&\alpha_{1}&0\end{pmatrix},$ where
$\mathrm{c}=(\alpha_{1},\alpha_{4})\in\mathbb{F}^{2}$ and $\alpha_{4}\neq 0$

  * •

$A_{5,2}(1,0)=\begin{pmatrix}1&0&0&0\\\ 1&0&1&0\end{pmatrix},$

  * •

$A_{6,2}(\mathrm{c})=\begin{pmatrix}\alpha_{1}&0&0&\alpha_{4}\\\
0&1+\alpha_{1}&\alpha_{1}&0\end{pmatrix}\simeq\begin{pmatrix}\alpha_{1}&0&0&a^{2}\alpha_{4}\\\
0&1+\alpha_{1}&\alpha_{1}&0\end{pmatrix},$ where
$\mathrm{c}=(\alpha_{1},\alpha_{4})\in\mathbb{F}^{2},$ $a\in\mathbb{F}$ and
$a\neq 0$

  * •

$A_{7,2}(\mathrm{c})=\begin{pmatrix}\alpha_{1}&1&1&0\\\
\beta_{1}&1+\alpha_{1}&\alpha_{1}&1\end{pmatrix}\simeq\begin{pmatrix}\alpha_{1}&1&1&0\\\
\beta_{1}+a\alpha_{1}+a+a^{2}&1+\alpha_{1}&\alpha_{1}&1\end{pmatrix},$ where
$\mathrm{c}=(\alpha_{1},\beta_{1})\in\mathbb{F}^{2}$ and $a\in\mathbb{F}$

  * •

$A_{8,2}(\mathrm{c})=\begin{pmatrix}0&1&1&1\\\
\beta_{1}&0&0&1\end{pmatrix}\simeq\begin{pmatrix}0&1&1&1\\\
\beta_{1}^{{}^{\prime}}(a)&0&0&1\end{pmatrix},$ where the polynomial
$(\beta_{1}t^{3}+t+1)(\beta_{1}t^{2}+\beta_{1}t+1)$ has no root in
$\mathbb{F},$ $a\in\mathbb{F}$ and
$\beta_{1}^{{}^{\prime}}(t)=\frac{(\beta_{1}^{2}t^{3}+\beta_{1}t+\beta_{1})^{2}}{(\beta_{1}t^{2}+\beta_{1}t+1)^{3}}$

  * •

$A_{9,2}(\mathrm{c})=\begin{pmatrix}0&0&0&1\\\
\beta_{1}&0&0&0\end{pmatrix}\simeq\begin{pmatrix}0&0&0&1\\\
a^{3}\beta_{1}^{\pm 1}&0&0&0\end{pmatrix},$ where
$a,\mathrm{c}=\beta_{1}\in\mathbb{F}$ and $a\neq 0,$

the polynomial $\beta_{1}+t^{3}$ has no root in $\mathbb{F}$

  * •

$A_{10,2}(\mathrm{c})=\begin{pmatrix}1&1&1&0\\\
\beta_{1}&1&1&1\end{pmatrix}\simeq\begin{pmatrix}1&1&1&0\\\
\beta_{1}+a+a^{2}&1&1&1\end{pmatrix},$ where
$a,\mathrm{c}=\beta_{1}\in\mathbb{F}$

  * •

$A_{11,2}(\mathrm{c})=\begin{pmatrix}0&1&1&0\\\ \beta_{1}&0&0&1\\\
\end{pmatrix}\simeq\begin{pmatrix}0&1&1&0\\\ b^{2}(\beta_{1}+a^{2})&0&0&1\\\
\end{pmatrix},$ where $a,b\in\mathbb{F}$ and $b\neq 0$

  * •

$A_{12,2}=\begin{pmatrix}0&0&0&0\\\ 1&0&0&0\\\ \end{pmatrix}$

######  Theorem 3.

Any non-trivial $2$-dimensional algebra over a field $\mathbb{F}$
$(Char(\mathbb{F})=3)$ is isomorphic to only one of the following, listed by
their matrices of structure constants, such algebras:

  * •

$A_{1,3}(\mathrm{c})=\begin{pmatrix}\alpha_{1}&\alpha_{2}&\alpha_{2}+1&\alpha_{4}\\\
\beta_{1}&-\alpha_{1}&1-\alpha_{1}&-\alpha_{2}\end{pmatrix},$ where
$\mathrm{c}=(\alpha_{1},\alpha_{2},\alpha_{4},\beta_{1})\in\mathbb{F}^{4}$

  * •

$A_{2,3}(\mathrm{c})=\begin{pmatrix}\alpha_{1}&0&0&\alpha_{4}\\\
1&\beta_{2}&1-\alpha_{1}&0\end{pmatrix},$ where
$\mathrm{c}=(\alpha_{1},\alpha_{4},\beta_{2})\in\mathbb{F}^{3},$ and
$\alpha_{4}\neq 0$

  * •

$A_{3,3}(\mathrm{c})=\begin{pmatrix}\alpha_{1}&0&0&\alpha_{4}\\\
0&\beta_{2}&1-\alpha_{1}&0\end{pmatrix}\simeq\begin{pmatrix}\alpha_{1}&0&0&a^{2}\alpha_{4}\\\
0&\beta_{2}&1-\alpha_{1}&0\end{pmatrix},$ where
$\mathrm{c}=(\alpha_{1},\alpha_{4},\beta_{2})\in\mathbb{F}^{3},$
$a\in\mathbb{F}$ and $a\neq 0$

  * •

$A_{4,3}(\mathrm{c})=\begin{pmatrix}0&1&1&0\\\
\beta_{1}&\beta_{2}&1&-1\end{pmatrix},$ where
$\mathrm{c}=(\beta_{1},\beta_{2})\in\mathbb{F}^{2}$

  * •

$A_{5,3}(\mathrm{c})=\begin{pmatrix}\alpha_{1}&0&0&0\\\
1&2\alpha_{1}-1&1-\alpha_{1}&0\end{pmatrix},$ where
$\mathrm{c}=\alpha_{1}\in\mathbb{F}$

  * •

$A_{6,3}(\mathrm{c})=\begin{pmatrix}\alpha_{1}&0&0&\alpha_{4}\\\
1&1-\alpha_{1}&-\alpha_{1}&0\end{pmatrix},$ where
$\mathrm{c}=(\alpha_{1},\alpha_{4})\in\mathbb{F}^{2}$ and $\alpha_{4}\neq 0$

  * •

$A_{7,3}(\mathrm{c})=\begin{pmatrix}\alpha_{1}&0&0&\alpha_{4}\\\
0&1-\alpha_{1}&-\alpha_{1}&0\end{pmatrix}\simeq\begin{pmatrix}\alpha_{1}&0&0&a^{2}\alpha_{4}\\\
0&1-\alpha_{1}&-\alpha_{1}&0\end{pmatrix},$ where
$\mathrm{c}=(\alpha_{1},\alpha_{4})\in\mathbb{F}^{2},$ $a\in\mathbb{F}$ and
$a\neq 0$

  * •

$A_{8,3}(\mathrm{c})=\begin{pmatrix}0&1&1&0\\\ \beta_{1}&1&0&-1\end{pmatrix},$
where $\mathrm{c}=\beta_{1}\in\mathbb{F}$

  * •

$A_{9,3}(\beta_{1})=\begin{pmatrix}0&1&1&1\\\
\beta_{1}&0&0&-1\end{pmatrix}\simeq\begin{pmatrix}0&1&1&1\\\
\beta_{1}^{\prime}(a)&0&0&-1\end{pmatrix},$ where the polynomial
$(\beta_{1}-t^{3})(\beta_{1}t^{2}+\beta_{1}t+1)(\beta_{1}^{2}t^{3}+\beta_{1}-2)$
has no root in $\mathbb{F},$ $a\in\mathbb{F}$ and
$\beta_{1}^{{}^{\prime}}(t)=\frac{(\beta_{1}^{2}t^{3}+\beta_{1}-2)^{2}}{(\beta_{1}t^{2}+\beta_{1}t+1)^{3}}$

  * •

$A_{10,3}(\mathrm{c})=\begin{pmatrix}0&0&0&1\\\
\beta_{1}&0&0&0\end{pmatrix}\simeq\begin{pmatrix}0&0&0&1\\\
a^{3}\beta_{1}^{\pm 1}&0&0&0\end{pmatrix},$
where the polynomial $\beta_{1}-t^{3}$ has no root,
$a,\mathrm{c}=\beta_{1}\in\mathbb{F}$ and $a,\beta_{1}\neq 0$

  * •

$A_{11,3}(\mathrm{c})=\begin{pmatrix}0&1&1&0\\\ \beta_{1}&0&0&-1\\\
\end{pmatrix}\simeq\begin{pmatrix}0&1&1&0\\\ a^{2}\beta_{1}&0&0&-1\\\
\end{pmatrix},$ where $a,\mathrm{c}=\beta_{1}\in\mathbb{F},$ $a\neq 0$

  * •

$A_{12,3}=\begin{pmatrix}1&0&0&0\\\ 1&-1&-1&0\end{pmatrix},$

  * •

$A_{13,3}=\begin{pmatrix}0&0&0&0\\\ 1&0&0&0\end{pmatrix}.$

In the next sections we classify, up to isomorphism, various classes of two-
dimensional algebras over any field relying on these results.

##  3\. A complete classification of $2$-dimensional endo-commutative algebras

######  Definition 1.

An algebra $\mathbf{A}$ is said to be endo-commutative if
$\mathrm{x}^{2}\mathrm{y}^{2}=(\mathrm{x}\mathrm{y})^{2},$ for any
$\mathrm{x},\mathrm{y}\in\mathbf{A}.$

According to (2.1) we write

$\mathrm{x}^{2}=\mathbf{e}Ax^{\otimes 2},\ \
\mathrm{y}^{2}=\mathbf{e}Ay^{\otimes 2},\ \
\mathrm{x}^{2}\mathrm{y}^{2}=\mathbf{e}A(Ax^{\otimes 2}\otimes Ay^{\otimes
2})=\mathbf{e}A(A\otimes A)(x^{\otimes 2}\otimes y^{\otimes 2})$

and

$(\mathrm{x}\mathrm{y})^{2}=\mathbf{e}A(A(x\otimes y)\otimes A(x\otimes
y))=\mathbf{e}A(A\otimes A)(x\otimes y)^{\otimes 2}.$

Therefore, the equality
$\mathrm{x}^{2}\mathrm{y}^{2}=(\mathrm{x}\mathrm{y})^{2}$ in terms of MSC and
the coordinate vectors is written as follows

$\mathbf{e}A(A\otimes A)(x^{\otimes 2}\otimes y^{\otimes
2})=\mathbf{e}A(A\otimes A)(x\otimes y)^{\otimes 2}$

and

$A(A\otimes A)(x^{\otimes 2}\otimes y^{\otimes 2}-(x\otimes y)^{\otimes 2})=0$

or just

(3.1) $A(A\otimes A)(x\otimes x\otimes y\otimes y-x\otimes y\otimes x\otimes
y)=0.$

Let now $\mathbf{A}$ be a two-dimensional algebra over a field $\mathbb{F}$
and let

$A=\left(\begin{array}[]{cccc}\alpha_{1}&\alpha_{2}&\alpha_{3}&\alpha_{4}\\\
\beta_{1}&\beta_{2}&\beta_{3}&\beta_{4}\end{array}\right)$

be its MSC on a basis $\mathbf{e}=\\{\mathrm{e}_{1},\mathrm{e}_{2}\\}$.

Then (3.1) is nothing but

(3.2)
$\left\\{\begin{array}[]{llll}x_{1}^{2}y_{1}y_{2}(A_{3}-A_{5})+x_{1}^{2}y_{2}^{2}(A_{4}-A_{6})+x_{1}x_{2}y_{1}y_{2}(A_{6}-A_{4}+A_{11}-A_{13})\\\
\hfill+x_{1}x_{2}y_{1}^{2}(A_{5}-A_{3})+x_{1}x_{2}y_{2}^{2}(A_{12}-A_{14})+x_{2}^{2}y_{1}^{2}(A_{13}-A_{11})+x_{2}^{2}y_{1}y_{2}(A_{14}-A_{12})&=&0\\\
x_{1}^{2}y_{1}y_{2}(B_{3}-B_{5})+x_{1}^{2}y_{2}^{2}(B_{4}-B_{6})+x_{1}x_{2}y_{1}y_{2}(B_{6}-B_{4}+B_{11}-B_{13})\\\
\hfill+x_{1}x_{2}y_{1}^{2}(B_{5}-B_{3})+x_{1}x_{2}y_{2}^{2}(B_{12}-B_{14})+x_{2}^{2}y_{1}^{2}(B_{13}-B_{11})+x_{2}^{2}y_{1}y_{2}(B_{14}-B_{12})&=&0,\end{array}\right.$

where

$\begin{array}[]{rlll}A_{3}&=&\alpha_{1}^{2}\alpha_{3}+\alpha_{1}\alpha_{2}\beta_{3}+\alpha_{3}^{2}\beta_{1}+\alpha_{4}\beta_{1}\beta_{3},\\\
A_{4}&=&\alpha_{1}^{2}\alpha_{4}+\alpha_{1}\alpha_{2}\beta_{4}+\alpha_{3}\alpha_{4}\beta_{1}+\alpha_{4}\beta_{1}\beta_{4},\\\
A_{5}&=&\alpha_{1}^{2}\alpha_{2}+\alpha_{2}^{2}\beta_{1}+\alpha_{1}\alpha_{3}\beta_{2}+\alpha_{4}\beta_{1}\beta_{2},\\\
A_{6}&=&\alpha_{1}\alpha_{2}^{2}+\alpha_{2}^{2}\beta_{2}+\alpha_{2}\alpha_{3}\beta_{2}+\alpha_{4}\beta_{2}^{2},\\\
A_{11}&=&\alpha_{1}\alpha_{3}^{2}+\alpha_{2}\alpha_{3}\beta_{3}+\alpha_{3}^{2}\beta_{3}+\alpha_{4}\beta_{3}^{2},\\\
A_{12}&=&\alpha_{1}\alpha_{3}\alpha_{4}+\alpha_{2}\alpha_{3}\beta_{4}+\alpha_{3}\alpha_{4}\beta_{3}+\alpha_{4}\beta_{3}\beta_{4},\\\
A_{13}&=&\alpha_{1}^{2}\alpha_{4}+\alpha_{2}\alpha_{4}\beta_{1}+\alpha_{1}\alpha_{3}\beta_{4}+\alpha_{4}\beta_{1}\beta_{4},\\\
A_{14}&=&\alpha_{1}\alpha_{2}\alpha_{4}+\alpha_{2}\alpha_{4}\beta_{2}+\alpha_{2}\alpha_{3}\beta_{4}+\alpha_{4}\beta_{2}\beta_{4},\\\
B_{3}&=&\alpha_{1}\alpha_{3}\beta_{1}+\alpha_{1}\beta_{2}\beta_{3}+\alpha_{3}\beta_{1}\beta_{3}+\beta_{1}\beta_{3}\beta_{4},\\\
B_{4}&=&\alpha_{1}\alpha_{4}\beta_{1}+\alpha_{1}\beta_{2}\beta_{4}+\alpha_{4}\beta_{1}\beta_{3}+\beta_{1}\beta_{4}^{2},\\\
B_{5}&=&\alpha_{1}\alpha_{2}\beta_{1}+\alpha_{2}\beta_{1}\beta_{2}+\alpha_{1}\beta_{2}\beta_{3}+\beta_{1}\beta_{2}\beta_{4},\\\
B_{6}&=&\alpha_{2}^{2}\beta_{1}+\alpha_{2}\beta_{2}^{2}+\alpha_{2}\beta_{2}\beta_{3}+\beta_{2}^{2}\beta_{4},\\\
B_{11}&=&\alpha_{3}^{2}\beta_{1}+\alpha_{3}\beta_{2}\beta_{3}+\alpha_{3}\beta_{3}^{2}+\beta_{3}^{2}\beta_{4},\\\
B_{12}&=&\alpha_{3}\alpha_{4}\beta_{1}+\alpha_{3}\beta_{2}\beta_{4}+\alpha_{4}\beta_{3}^{2}+\beta_{3}\beta_{4}^{2},\\\
B_{13}&=&\alpha_{1}\alpha_{4}\beta_{1}+\alpha_{4}\beta_{1}\beta_{2}+\alpha_{1}\beta_{3}\beta_{4}+\beta_{1}\beta_{4}^{2},\\\
B_{14}&=&\alpha_{2}\alpha_{4}\beta_{1}+\alpha_{4}\beta_{2}^{2}+\alpha_{2}\beta_{3}\beta_{4}+\beta_{2}\beta_{4}^{2}.\end{array}$

Note that the set of functions
$\\{x_{1}^{2}y_{1}y_{2},x_{1}^{2}y_{2}^{2},x_{1}x_{2}y_{1}^{2},x_{1}x_{2}y_{1}y_{2},x_{1}x_{2}y_{2}^{2},x_{2}^{2}y_{1}^{2},x_{2}^{2}y_{1}y_{2}\\}$
is linearly independent. Therefore, the system (3.2) in terms of $A_{i}$ and
$B_{j}$ ($i,j=3,4,...,13$) can be rewritten as follows

(3.3) $\left\\{\begin{array}[]{ccc}A_{3}-A_{5}&=&0\\\ A_{4}-A_{6}&=&0\\\
A_{6}-A_{4}+A_{11}-A_{13}&=&0\\\ A_{12}-A_{14}&=&0\\\ A_{13}-A_{11}&=&0\\\
B_{3}-B_{5}&=&0\\\ B_{4}-B_{6}&=&0\\\ B_{6}-B_{4}+B_{11}-B_{13}&=&0\\\
B_{12}-B_{14}&=&0\\\ B_{13}-B_{11}&=&0\\\ \end{array}\right.$

The equations 3 and 8 in the system can be removed since they are linearly
depend on the remaining equations.

In terms of the structure constants $\alpha_{i},\beta_{j}$ ($i,j=1,2,...,4$)
the system (3.3) is written as follows

(3.4)
$\left\\{\begin{array}[]{lccc}(\alpha_{1}^{2}\alpha_{3}+\alpha_{1}\alpha_{2}\beta_{3}+\alpha_{3}^{2}\beta_{1}+\alpha_{4}\beta_{1}\beta_{3})-(\alpha_{1}^{2}\alpha_{2}+\alpha_{2}^{2}\beta_{1}+\alpha_{1}\alpha_{3}\beta_{2}+\alpha_{4}\beta_{1}\beta_{2})&=&0\\\
(\alpha_{1}^{2}\alpha_{4}+\alpha_{1}\alpha_{2}\beta_{4}+\alpha_{3}\alpha_{4}\beta_{1}+\alpha_{4}\beta_{1}\beta_{4})-(\alpha_{1}\alpha_{2}^{2}+\alpha_{2}^{2}\beta_{2}+\alpha_{2}\alpha_{3}\beta_{2}+\alpha_{4}\beta_{2}^{2})&=&0\\\
(\alpha_{1}\alpha_{3}\alpha_{4}+\alpha_{2}\alpha_{3}\beta_{4}+\alpha_{3}\alpha_{4}\beta_{3}+\alpha_{4}\beta_{3}\beta_{4})-(\alpha_{1}\alpha_{2}\alpha_{4}+\alpha_{2}\alpha_{4}\beta_{2}+\alpha_{2}\alpha_{3}\beta_{4}+\alpha_{4}\beta_{2}\beta_{4})&=&0\\\
(\alpha_{1}^{2}\alpha_{4}+\alpha_{2}\alpha_{4}\beta_{1}+\alpha_{1}\alpha_{3}\beta_{4}+\alpha_{4}\beta_{1}\beta_{4})-(\alpha_{1}\alpha_{3}^{2}+\alpha_{2}\alpha_{3}\beta_{3}+\alpha_{3}^{2}\beta_{3}+\alpha_{4}\beta_{3}^{2})&=&0\\\
(\alpha_{1}\alpha_{3}\beta_{1}+\alpha_{1}\beta_{2}\beta_{3}+\alpha_{3}\beta_{1}\beta_{3}+\beta_{1}\beta_{3}\beta_{4})-(\alpha_{1}\alpha_{2}\beta_{1}+\alpha_{2}\beta_{1}\beta_{2}+\alpha_{1}\beta_{2}\beta_{3}+\beta_{1}\beta_{2}\beta_{4})&=&0\\\
(\alpha_{1}\alpha_{4}\beta_{1}+\alpha_{1}\beta_{2}\beta_{4}+\alpha_{4}\beta_{1}\beta_{3}+\beta_{1}\beta_{4}^{2})-(\alpha_{2}^{2}\beta_{1}+\alpha_{2}\beta_{2}^{2}+\alpha_{2}\beta_{2}\beta_{3}+\beta_{2}^{2}\beta_{4})&=&0\\\
(\alpha_{3}\alpha_{4}\beta_{1}+\alpha_{3}\beta_{2}\beta_{4}+\alpha_{4}\beta_{3}^{2}+\beta_{3}\beta_{4}^{2})-(\alpha_{2}\alpha_{4}\beta_{1}+\alpha_{4}\beta_{2}^{2}+\alpha_{2}\beta_{3}\beta_{4}+\beta_{2}\beta_{4}^{2})&=&0\\\
(\alpha_{1}\alpha_{4}\beta_{1}+\alpha_{4}\beta_{1}\beta_{2}+\alpha_{1}\beta_{3}\beta_{4}+\beta_{1}\beta_{4}^{2})-(\alpha_{3}^{2}\beta_{1}+\alpha_{3}\beta_{2}\beta_{3}+\alpha_{3}\beta_{3}^{2}+\beta_{3}^{2}\beta_{4})&=&0\\\
\end{array}\right.$

###  3.1. Classification over $\mathbb{F}$ with $\mathbf{Char(\mathbb{F})\neq
2,3}$

__

It is clear that in this case to get classification of two-dimensional endo-
commutative algebras it suffices to solve the general system (3.4) for each
MSC $A_{i}$ of the algebras $\mathbf{A}_{i},$ ($i=1,2,...,13$) given in
Theorem 1.

For
$A=A_{1}(\mathrm{c})=\begin{pmatrix}\alpha_{1}&\alpha_{2}&1+\alpha_{2}&\alpha_{4}\\\
\beta_{1}&-\alpha_{1}&1-\alpha_{1}&-\alpha_{2}\end{pmatrix},$ where
$\mathrm{c}=(\alpha_{1},\alpha_{2},\alpha_{4},\beta_{1})\in\mathbb{F}^{4},$
the system of equations (3.4) looks like

$\left\\{\begin{array}[]{cc}(\alpha_{1}^{2}(1+\alpha_{2})+\alpha_{1}\alpha_{2}(1-\alpha_{1})+(1+\alpha_{2})^{2}\beta_{1}+\alpha_{4}\beta_{1}(1-\alpha_{1}))-(\alpha_{1}^{2}\alpha_{2}+\alpha_{2}^{2}\beta_{1}\\\
-\alpha_{1}^{2}(1+\alpha_{2})-\alpha_{1}\alpha_{4}\beta_{1})=0\\\
(\alpha_{1}^{2}\alpha_{4}-\alpha_{1}\alpha_{2}^{2}+(1+\alpha_{2})\alpha_{4}\beta_{1}-\alpha_{2}\alpha_{4}\beta_{1})-(\alpha_{1}\alpha_{2}^{2}-\alpha_{1}\alpha_{2}^{2}\\\
-\alpha_{1}\alpha_{2}(1+\alpha_{2})+\alpha_{4}(-\alpha_{1})^{2})=0\\\
(\alpha_{1}(1+\alpha_{2})\alpha_{4}+(1+\alpha_{2})\alpha_{4}(1-\alpha_{1})-\alpha_{2}\alpha_{4}(1-\alpha_{1}))-(\alpha_{1}\alpha_{2}\alpha_{4}-\alpha_{1}\alpha_{2}\alpha_{4}\\\
+\alpha_{4}(-\alpha_{1})(-\alpha_{2}))=0\\\
(\alpha_{1}^{2}\alpha_{4}+\alpha_{2}\alpha_{4}\beta_{1}+\alpha_{1}(1+\alpha_{2})(-\alpha_{2})+\alpha_{4}\beta_{1}(-\alpha_{2}))-(\alpha_{1}(1+\alpha_{2})^{2}\\\
+\alpha_{2}(1+\alpha_{2})(1-\alpha_{1})+(1+\alpha_{2})^{2}(1-\alpha_{1})+\alpha_{4}(1-\alpha_{1})^{2})=0\\\
(\alpha_{1}(1+\alpha_{2})\beta_{1}+(1+\alpha_{2})\beta_{1}(1-\alpha_{1})+\beta_{1}(1-\alpha_{1})(-\alpha_{2}))-(\alpha_{1}\alpha_{2}\beta_{1}-\alpha_{1}\alpha_{2}\beta_{1}\\\
+\beta_{1}(-\alpha_{1})(-\alpha_{2}))=0\\\
(\alpha_{1}\alpha_{4}\beta_{1}-\alpha_{1}\alpha_{2}\beta_{2}+\alpha_{4}\beta_{1}(1-\alpha_{1})+\beta_{1}(-\alpha_{2})^{2})-(\alpha_{2}^{2}\beta_{1}+\alpha_{2}(-\alpha_{1})^{2}\\\
+\alpha_{2}(-\alpha_{1})(1-\alpha_{1})+(-\alpha_{1})^{2}(-\alpha_{2})=0\\\
((1+\alpha_{2})\alpha_{4}\beta_{1}+(1+\alpha_{2})(-\alpha_{1})(-\alpha_{2})+\alpha_{4}(1-\alpha_{1})^{2}+(1-\alpha_{1})(-\alpha_{2})^{2})-(\alpha_{2}\alpha_{4}\beta_{1}\\\
+\alpha_{4}(-\alpha_{1})^{2}+\alpha_{2}(1-\alpha_{1})(-\alpha_{2})-\alpha_{1}(-\alpha_{2})^{2})=0\\\
(\alpha_{1}\alpha_{4}\beta_{1}-\alpha_{1}\alpha_{4}\beta_{1}+\alpha_{1}(1-\alpha_{1})(-\alpha_{2})+\beta_{1}(-\alpha_{2})^{2})-((1+\alpha_{2})^{2}\beta_{1}+(1+\alpha_{2})(-\alpha_{1})(1-\alpha_{1})\\\
+(1+\alpha_{2})(1-\alpha_{1})^{2}+(1-\alpha_{1})^{2}(-\alpha_{2}))=0\\\
\end{array}\right..$

Simplifying we get

$\left\\{\begin{array}[]{cc}2\alpha_{1}^{2}+\alpha_{1}\alpha_{2}+\beta_{1}+2\alpha_{2}\beta_{1}+\alpha_{4}\beta_{1}=0\\\
\alpha_{4}\beta_{1}+\alpha_{1}\alpha_{2}=0\\\ \alpha_{4}=0\\\
-3\alpha_{2}-2\alpha_{2}^{2}-1-\alpha_{4}+2\alpha_{1}\alpha_{4}=0\\\
\beta_{1}=0\\\ \alpha_{4}\beta_{1}+\alpha_{1}\alpha_{2}=0\\\
\alpha_{4}\beta_{1}+\alpha_{1}\alpha_{2}+\alpha_{4}-2\alpha_{1}\alpha_{4}+2\alpha_{2}^{2}=0\\\
-\beta_{1}-2\alpha_{2}\beta_{1}+3\alpha_{1}-2\alpha_{1}^{2}-1=0\\\
\end{array}\right.$

Taking into account the 3rd and 5th equations we obtain

$\left\\{\begin{array}[]{cc}2\alpha_{1}^{2}+\alpha_{1}\alpha_{2}=0\\\
\alpha_{1}\alpha_{2}=0\\\ \alpha_{4}=0\\\ -3\alpha_{2}-2\alpha_{2}^{2}-1=0\\\
\beta_{1}=0\\\ \alpha_{1}\alpha_{2}=0\\\
\alpha_{1}\alpha_{2}+2\alpha_{2}^{2}=0\\\ 3\alpha_{1}-2\alpha_{1}^{2}-1=0\\\
\end{array}\right.$

As far as the 2${}^{\mathrm{nd}},$ 4th and 8th equations show inconsistence of
the system, so there is no endo-commutative algebra in this class of algebras.

Similarly, if $A=A_{2}(\mathrm{c})=\begin{pmatrix}\alpha_{1}&0&0&\alpha_{4}\\\
1&\beta_{2}&1-\alpha_{1}&0\end{pmatrix},$ where
$\mathrm{c}=(\alpha_{1},\alpha_{4},\beta_{2})\in\mathbb{F}^{3},$ then one gets

$\left\\{\begin{array}[]{cc}(1-\alpha_{1})\alpha_{4}-\alpha_{4}\beta_{2}=0\\\
\alpha_{1}^{2}\alpha_{4}-\alpha_{4}\beta_{2}^{2}=0\\\
\alpha_{1}^{2}\alpha_{4}-(1-\alpha_{1})^{2}\alpha_{4}=0\\\ \alpha_{4}=0\\\
(1-\alpha_{1})^{2}\alpha_{4}+\alpha_{4}\beta_{2}^{2}=0\\\
\alpha_{1}\alpha_{4}+\alpha_{4}\beta_{2}=0\\\ \end{array}\right.$

for which $(\alpha_{1},0,\beta_{2})$ is the only solution.

If $A=A_{3}(\mathrm{c})=\begin{pmatrix}\alpha_{1}&0&0&\alpha_{4}\\\
0&\beta_{2}&1-\alpha_{1}&0\end{pmatrix},$ where
$\mathrm{c}=(\alpha_{1},\alpha_{4},\beta_{2})\in\mathbb{F}^{3},$ then the
system of equations looks like

$\left\\{\begin{array}[]{cc}\alpha_{1}^{2}\alpha_{4}-\alpha_{4}\beta_{2}^{2}=0\\\
\alpha_{1}^{2}\alpha_{4}-(1-\alpha_{1})^{2}\alpha_{4}=0\\\
(1-\alpha_{1})^{2}\alpha_{4}-\alpha_{4}\beta_{2}^{2}=0\\\ \end{array}\right.$

which has only the following solutions : $(\alpha_{1},0,\beta_{2})$ and
$(\frac{1}{2},\alpha_{4},\frac{1}{2})$.

In $A=A_{4}(\mathrm{c})=\begin{pmatrix}0&1&1&0\\\
\beta_{1}&\beta_{2}&1&-1\end{pmatrix},$ where
$\mathrm{c}=(\beta_{1},\beta_{2})\in\mathbb{F}^{2},$ case the corresponding
system

$\left\\{\begin{array}[]{cc}2\beta_{2}=0\\\ 2=0\\\ -\beta_{2}=0\\\
-2\beta_{2}+2=0\\\ \end{array}\right.$

is obviously inconsistent.

If $A=A_{5}(\mathrm{c})=\begin{pmatrix}\alpha_{1}&0&0&0\\\
1&2\alpha_{1}-1&1-\alpha_{1}&0\end{pmatrix},$ where
$\mathrm{c}=\alpha_{1}\in\mathbb{F},$ the corresponding system consists of
identities, i.e., all the algebras in this class are endo-commutative.

In $A=A_{6}(\mathrm{c})=\begin{pmatrix}\alpha_{1}&0&0&\alpha_{4}\\\
1&1-\alpha_{1}&-\alpha_{1}&0\end{pmatrix},$ where
$\mathrm{c}=(\alpha_{1},\alpha_{4})\in\mathbb{F}^{2},$ case the system is

$\left\\{\begin{array}[]{cc}\alpha_{1}^{2}\alpha_{4}-\alpha_{4}(1-\alpha_{1})^{2}=0\\\
\alpha_{4}(-\alpha_{1})^{2}-\alpha_{4}(1-\alpha_{1})^{2}=0\\\ \alpha_{4}=0\\\
\end{array}\right.$

The solution to the system is $(\alpha_{1},0),$ where
$\alpha_{1}\in\mathbb{F}.$

If $A=A_{7}(\mathrm{c})=\begin{pmatrix}\alpha_{1}&0&0&\alpha_{4}\\\
0&1-\alpha_{1}&-\alpha_{1}&0\end{pmatrix},\ \mbox{where}\
\mathrm{c}=(\alpha_{1},\alpha_{4})\in\mathbb{F}^{2}$ then the system becomes

$\left\\{\begin{array}[]{cc}\alpha_{1}^{2}\alpha_{4}-\alpha_{4}(1-\alpha_{1})^{2}=0\\\
\alpha_{1}^{2}\alpha_{4}-\alpha_{4}(-\alpha_{1})^{2}=0\\\
\alpha_{4}(-\alpha_{1})^{2}-\alpha_{4}(1-\alpha_{1})^{2}=0.\\\
\end{array}\right.$

Obviously, $(\alpha_{1},0)\ \mbox{and}\ \left(\frac{1}{2},\alpha_{4}\right),\
\mbox{where}\ \alpha_{4}\neq 0$ are solutions to the system.

Let

$A=A_{8}(\mathrm{c})=\begin{pmatrix}0&1&1&0\\\ \beta_{1}&1&0&-1\end{pmatrix},$

where $\mathrm{c}=\beta_{1}\in\mathbb{F}$. It is easy to see here that the
system is inconsistent.

Verifying the system of equations (3.4) for $A_{9},$ $A_{10}(\mathrm{c}),$
$A_{11}(\mathrm{c}),$ $A_{12}(\mathrm{c})$ and $A_{13}$ we get identities,
i.e., all these algebras happen to be endo-commutative algebras.

The final result for algebras over a field $\mathbb{F}$ $(Char\mathbb{F}\neq
2,3)$ is given as follows.

######  Theorem 4.

Any non-trivial $2$-dimensional endo-commutative algebra over a field
$\mathbb{F},$ $(Char(\mathbb{F})\neq 2,3)$ is isomorphic to only one of the
following listed by their matrices of structure constants, such algebras:

  * •

$A_{2}(\alpha_{1},0,\beta_{2})=\begin{pmatrix}\alpha_{1}&0&0&0\\\
1&\beta_{2}&1-\alpha_{1}&0\end{pmatrix},$ where
$\alpha_{1},\beta_{2}\in\mathbb{F}$

  * •

$A_{3}(\alpha_{1},0,\beta_{2})=\begin{pmatrix}\alpha_{1}&0&0&0\\\
0&\beta_{2}&1-\alpha_{1}&0\end{pmatrix},$ where
$\alpha_{1},\beta_{2}\in\mathbb{F}$

  * •

$A_{3}(\frac{1}{2},\alpha_{4},\frac{1}{2})=\begin{pmatrix}\frac{1}{2}&0&0&\alpha_{4}\\\
0&\frac{1}{2}&\frac{1}{2}&0\end{pmatrix},$ where $\alpha_{4}\in\mathbb{F}$ and
$\alpha_{4}\neq 0$

  * •

$A_{5}(\alpha_{1})=\begin{pmatrix}\alpha_{1}&0&0&0\\\
1&2\alpha_{1}-1&1-\alpha_{1}&0\end{pmatrix},$ where $\alpha_{1}\in\mathbb{F}$

  * •

$A_{6}(\alpha_{1},0)=\begin{pmatrix}\alpha_{1}&0&0&0\\\
1&1-\alpha_{1}&-\alpha_{1}&0\end{pmatrix},$ where $\alpha_{1}\in\mathbb{F}$

  * •

$A_{7}(\alpha_{1},0)=\begin{pmatrix}\alpha_{1}&0&0&0\\\
0&1-\alpha_{1}&-\alpha_{1}&0\end{pmatrix},$ where $\alpha_{1}\in\mathbb{F}$

  * •

$A_{7}(\frac{1}{2},\alpha_{4})=\begin{pmatrix}\frac{1}{2}&0&0&\alpha_{4}\\\
0&\frac{1}{2}&-\frac{1}{2}&0\end{pmatrix},$ where $\alpha_{4}\in\mathbb{F}$
and $\alpha_{4}\neq 0$

  * •

$A_{9}=\begin{pmatrix}\frac{1}{3}&0&0&0\\\
1&\frac{2}{3}&-\frac{1}{3}&0\end{pmatrix},$

  * •

$A_{10}(\beta_{1})=\begin{pmatrix}0&1&1&1\\\
\beta_{1}&0&0&-1\end{pmatrix}\simeq\begin{pmatrix}0&1&1&1\\\
\beta_{1}^{{}^{\prime}}(a)&0&0&-1\end{pmatrix},$
where $a,\beta_{1}\in\mathbb{F},$ the polynomial
$(\beta_{1}t^{3}-3t-1)(\beta_{1}t^{2}+\beta_{1}t+1)(\beta_{1}^{2}t^{3}+6\beta_{1}t^{2}+3\beta_{1}t+\beta_{1}-2)$
has no root in $\mathbb{F},$ and
$\beta_{1}^{{}^{\prime}}(t)=\frac{(\beta_{1}^{2}t^{3}+6\beta_{1}t^{2}+3\beta_{1}t+\beta_{1}-2)^{2}}{(\beta_{1}t^{2}+\beta_{1}t+1)^{3}}$

  * •

$A_{11}(\beta_{1})=\begin{pmatrix}0&0&0&1\\\ \beta_{1}^{\pm 1}&0&0&0\\\
\end{pmatrix}\simeq\begin{pmatrix}0&0&0&1\\\ a^{3}\beta_{1}^{\pm 1}&0&0&0\\\
\end{pmatrix},$
where the polynomial $\beta_{1}-t^{3}$ has no root in $\mathbb{F},$
$a,\beta_{1}\in\mathbb{F}$ and $a\neq 0$

  * •

$A_{12}(\beta_{1})=\begin{pmatrix}0&1&1&0\\\
\beta_{1}&0&0&-1\end{pmatrix}\simeq\begin{pmatrix}0&1&1&0\\\
a^{2}\beta_{1}&0&0&-1\end{pmatrix},$
where $a,\beta_{1}\in\mathbb{F}$ and $a\neq 0$

  * •

$A_{13}=\begin{pmatrix}0&0&0&0\\\ 1&0&0&0\end{pmatrix}.$

###  3.2. Classification over $\mathbb{F}$ with
$\mathbf{Char(\mathbb{F})=2,3}$

__

The description of two-dimensional endo-commutative algebras over $\mathbb{F}$
for the cases $Char(\mathbb{F})=2$ and $Char(\mathbb{F})=3$ can be obtained by
using Theorems 2 and 3, respectively. The proof is similar to that of Theorem
4. Therefore, we give the corresponding results without proof.

######  Theorem 5.

Any non-trivial $2$-dimensional endo-commutative algebra over a field
$\mathbb{F}$ $(Char(\mathbb{F})=2)$ is isomorphic to only one of the
following, listed by their matrices of structure constants, such algebras:

  * •

$A_{2,2}(\alpha_{1},0,1)=\begin{pmatrix}\alpha_{1}&0&0&0\\\
1&1&1+\alpha_{1}&0\end{pmatrix},$ where $\alpha_{1}\in\mathbb{F}$

  * •

$A_{3,2}(\alpha_{1},0,\beta_{2})=\begin{pmatrix}\alpha_{1}&0&0&0\\\
0&\beta_{2}&1+\alpha_{1}&0\end{pmatrix},$ where
$\alpha_{1},\beta_{2}\in\mathbb{F}$

  * •

$A_{4,2}(\alpha_{1},\beta_{1},0)=\begin{pmatrix}\alpha_{1}&1&1&0\\\
\beta_{1}&0&1+\alpha_{1}&1\end{pmatrix}\simeq\begin{pmatrix}\alpha_{1}&1&1&0\\\
\beta_{1}+a+a^{2}&0&1+\alpha_{1}&1\end{pmatrix},$
where $a,\alpha_{1},\beta_{1}\in\mathbb{F}$

  * •

$A_{5,2}(1,0)=\begin{pmatrix}1&0&0&0\\\ 1&0&1&0\end{pmatrix},$

  * •

$A_{6,2}(\alpha_{1},0)=\begin{pmatrix}\alpha_{1}&0&0&0\\\
0&1+\alpha_{1}&\alpha_{1}&0\end{pmatrix},$ where $\alpha_{1}\in\mathbb{F}$

  * •

$A_{7,2}(0,\beta_{1})=\begin{pmatrix}0&1&1&0\\\
\beta_{1}&1&0&1\end{pmatrix}\simeq\begin{pmatrix}0&1&1&0\\\
\beta_{1}+a+a^{2}&1&0&1\end{pmatrix},$ where $\beta_{1},a\in\mathbb{F}$

  * •

$A_{8,2}(\beta_{1})=\begin{pmatrix}0&1&1&1\\\
\beta_{1}&0&0&1\end{pmatrix}\simeq\begin{pmatrix}0&1&1&1\\\
\beta_{1}^{{}^{\prime}}(a)&0&0&1\end{pmatrix},$ where the polynomial
$(\beta_{1}t^{3}+t+1)(\beta_{1}t^{2}+\beta_{1}t+1)$ has no root in
$\mathbb{F},$ $a\in\mathbb{F}$ and
$\beta_{1}^{{}^{\prime}}(t)=\frac{(\beta_{1}^{2}t^{3}+\beta_{1}t+\beta_{1})^{2}}{(\beta_{1}t^{2}+\beta_{1}t+1)^{3}}$

  * •

$A_{9,2}(\beta_{1})=\begin{pmatrix}0&0&0&1\\\
\beta_{1}&0&0&0\end{pmatrix}\simeq\begin{pmatrix}0&0&0&1\\\
a^{3}\beta_{1}^{\pm 1}&0&0&0\end{pmatrix},$ where $\beta_{1},a\in\mathbb{F},$
$a\neq 0$ and
the polynomial $\beta_{1}+t^{3}$ has no root in $\mathbb{F}$

  * •

$A_{10,2}(\beta_{1})=\begin{pmatrix}1&1&1&0\\\
\beta_{1}&1&1&1\end{pmatrix}\simeq\begin{pmatrix}1&1&1&0\\\
\beta_{1}+a+a^{2}&1&1&1\end{pmatrix},$ where $\beta_{1},a\in\mathbb{F}$

  * •

$A_{11,2}(\beta_{1})=\begin{pmatrix}0&1&1&0\\\ \beta_{1}&0&0&1\\\
\end{pmatrix}\simeq\begin{pmatrix}0&1&1&0\\\ b^{2}(\beta_{1}+a^{2})&0&0&1\\\
\end{pmatrix},$ where $a,b\in\mathbb{F}$ and $b\neq 0$

  * •

$A_{12,2}=\begin{pmatrix}0&0&0&0\\\ 1&0&0&0\end{pmatrix}.$

In $\mathbb{F}=\mathbb{Z}_{2}$ case the theorem above can be stated in a more
detailed form as follows.

######  Corollary 3.1.

Any non-trivial $2$-dimensional endo-commutative algebra over a field
$\mathbb{F}=\mathbb{Z}_{2},$ is isomorphic to only one of the following
listed, by their matrices of structure constants, such algebras:

  * •

$A_{2,2}(0,0,1)=\begin{pmatrix}0&0&0&0\\\ 1&1&1&0\end{pmatrix},$

  * •

$A_{2,2}(1,0,1)=\begin{pmatrix}1&0&0&0\\\ 1&1&0&0\end{pmatrix},$

  * •

$A_{3,2}(0,0,0)=\begin{pmatrix}0&0&0&0\\\ 0&0&1&0\end{pmatrix},$

  * •

$A_{3,2}(0,0,1)=\begin{pmatrix}0&0&0&0\\\ 0&1&1&0\end{pmatrix},$

  * •

$A_{3,2}(1,0,0)=\begin{pmatrix}1&0&0&0\\\ 0&0&0&0\end{pmatrix},$

  * •

$A_{3,2}(1,0,1)=\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\end{pmatrix},$

  * •

$A_{4,2}(0,0,0)=\begin{pmatrix}0&1&1&0\\\ 0&0&1&1\end{pmatrix},$

  * •

$A_{4,2}(0,1,0)=\begin{pmatrix}0&1&1&0\\\ 1&0&1&1\end{pmatrix},$

  * •

$A_{4,2}(1,0,0)=\begin{pmatrix}1&1&1&0\\\ 0&0&0&1\end{pmatrix},$

  * •

$A_{4,2}(1,1,0)=\begin{pmatrix}1&1&1&0\\\ 1&0&0&1\end{pmatrix},$

  * •

$A_{5,2}(1,0)=\begin{pmatrix}1&0&0&0\\\ 1&0&1&0\end{pmatrix},$

  * •

$A_{6,2}(0,0)=\begin{pmatrix}0&0&0&0\\\ 0&1&0&0\end{pmatrix},$

  * •

$A_{6,2}(1,0)=\begin{pmatrix}1&0&0&0\\\ 0&0&1&0\end{pmatrix},$

  * •

$A_{7,2}(0,0)=\begin{pmatrix}0&1&1&0\\\ 0&1&0&1\end{pmatrix},$

  * •

$A_{7,2}(0,1)=\begin{pmatrix}0&1&1&0\\\ 1&1&0&1\end{pmatrix},$

  * •

$A_{8,2}(1)=\begin{pmatrix}0&1&1&1\\\ 1&0&0&1\end{pmatrix},$

  * •

$A_{10,2}(0)=\begin{pmatrix}1&1&1&0\\\ 0&1&1&1\end{pmatrix},$

  * •

$A_{10,2}(1)=\begin{pmatrix}1&1&1&0\\\ 1&1&1&1\end{pmatrix},$

  * •

$A_{11,2}(0)=\begin{pmatrix}0&1&1&0\\\ 0&0&0&1\\\ \end{pmatrix},$

  * •

$A_{12,2}=\begin{pmatrix}0&0&0&0\\\ 1&0&0&0\end{pmatrix}.$

######  Theorem 6.

Any non-trivial two-dimensional endo-commutative algebra over a field
$\mathbb{F}$ $(Char(\mathbb{F})=3)$ is isomorphic to only one of the following
listed, by their matrices of structure constants, such algebras:

  * •

$A_{3,3}(\alpha_{1},0,\beta_{2})=\begin{pmatrix}\alpha_{1}&0&0&0\\\
0&\beta_{2}&1-\alpha_{1}&0\end{pmatrix},\ \mbox{where}\
\alpha_{1},\beta_{2}\in\mathbb{F}$

  * •

$A_{3,3}(2,\alpha_{4},1)=\begin{pmatrix}2&0&0&\alpha_{4}\\\
0&1&2&0\end{pmatrix}\simeq\begin{pmatrix}2&0&0&a^{2}\alpha_{4}\\\
0&1&2&0\end{pmatrix},\ \mbox{where}\ a,\alpha_{4}\in\mathbb{F}$ and
$a,\alpha_{4}\neq 0$

  * •

$A_{3,3}(2,\alpha_{4},2)=\begin{pmatrix}2&0&0&\alpha_{4}\\\
0&2&2&0\end{pmatrix}\simeq\begin{pmatrix}2&0&0&a^{2}\alpha_{4}\\\
0&2&2&0\end{pmatrix},\ \mbox{where}\ a,\alpha_{4}\in\mathbb{F}$ and
$a,\alpha_{4}\neq 0$

  * •

$A_{5,3}(\alpha_{1})=\begin{pmatrix}\alpha_{1}&0&0&0\\\
1&2\alpha_{1}-1&1-\alpha_{1}&0\end{pmatrix},\ \mbox{where}\
\alpha_{1}\in\mathbb{F}$

  * •

$A_{7,3}(\alpha_{1},0)=\begin{pmatrix}\alpha_{1}&0&0&0\\\
0&1-\alpha_{1}&-\alpha_{1}&0\end{pmatrix},\ \mbox{where}\
\alpha_{1}\in\mathbb{F}$

  * •

$A_{7,3}(2,\alpha_{4})=\begin{pmatrix}2&0&0&\alpha_{4}\\\
0&2&1&0\end{pmatrix}\simeq\begin{pmatrix}2&0&0&a^{2}\alpha_{4}\\\
0&2&1&0\end{pmatrix},$ where $a,\alpha_{4}\in\mathbb{F},$ and
$a,\alpha_{4}\neq 0$

  * •

$A_{9,3}(\beta_{1})=\begin{pmatrix}0&1&1&1\\\
\beta_{1}&0&0&-1\end{pmatrix}\simeq\begin{pmatrix}0&1&1&1\\\
\beta^{\prime}_{1}(a)&0&0&-1\end{pmatrix},$ where the polynomial
$(\beta_{1}-t^{3})(\beta_{1}t^{2}+\beta_{1}t+1)(\beta_{1}^{2}t^{3}+\beta_{1}-2)$
has no root in $\mathbb{F},$ $a\in\mathbb{F}$ and
$\beta_{1}^{{}^{\prime}}(t)=\frac{(\beta_{1}^{2}t^{3}+\beta_{1}-2)^{2}}{(\beta_{1}t^{2}+\beta_{1}t+1)^{3}}$

  * •

$A_{10,3}(\beta_{1})=\begin{pmatrix}0&0&0&1\\\
\beta_{1}&0&0&0\end{pmatrix}\simeq\begin{pmatrix}0&0&0&1\\\
a^{3}\beta_{1}^{\pm 1}&0&0&0\end{pmatrix},$ where polynomial $\beta_{1}-t^{3}$
has no root, $a,\beta_{1}\in\mathbb{F},$ and $a,\beta_{1}\neq 0$

  * •

$A_{11,3}(\beta_{1})=\begin{pmatrix}0&1&1&0\\\ \beta_{1}&0&0&-1\\\
\end{pmatrix}\simeq\begin{pmatrix}0&1&1&0\\\ a^{2}\beta_{1}&0&0&-1\\\
\end{pmatrix},\ \mbox{where}\ a,\beta_{1}\in\mathbb{F},$ and $a\neq 0,$

  * •

$A_{12,3}=\begin{pmatrix}1&0&0&0\\\ 1&-1&-1&0\end{pmatrix},$

  * •

$A_{13,3}=\begin{pmatrix}0&0&0&0\\\ 1&0&0&0\end{pmatrix}.$

##  4\. Two-dimensional endo-commutative curled algebras

In [26] the class of endo-commutative algebras was split into two: curled and
straight algebras. There one can find a list of such curled algebras, up to
isomorphism, over the field $\mathbb{F}=\mathbb{Z}_{2}$ as well. In this
section first we classify all two-dimensional curved algebras then by the use
of previous section results we give the classification of all endo-commutative
curled algebras, up to isomorphism, over any field $\mathbb{F}$.

######  Definition 2.

An algebra $\mathbf{A}$ over a field $\mathbb{F}$ is said to be curled if
$\mathrm{x}^{2}=\mathrm{\lambda(x)}\mathrm{x},$ for any
$\mathrm{x}\in\mathbf{A},$ where
$\mathrm{\lambda(x)}=\lambda(x_{1},x_{2})\in\mathbb{F}.$

Let $\mathbf{e}=\\{\mathrm{e}_{1},\mathrm{e}_{2},...,\mathrm{e}_{n}\\}$ be a
basis of $\mathbf{A}$. According to (2.1) one has
$\mathrm{x}^{2}=\mathbf{e}Ax^{\otimes 2},$
$\mathrm{\lambda(x)}\mathrm{x}=\mathbf{e}\lambda(x)x$ and therefore, the
equality $\mathrm{x}^{2}=\mathrm{\lambda(x)}\mathrm{x}$ can be written as
follows

(4.1) $Ax^{\otimes 2}-\lambda(x)x=0$

Consider two-dimensional case. One has
$\mathbf{e}=\\{\mathrm{e}_{1},\mathrm{e}_{2}\\}$ and

$A=\left(\begin{array}[]{cccc}\alpha_{1}^{{}^{\prime}}&\alpha_{2}^{{}^{\prime}}&\alpha_{3}^{{}^{\prime}}&\alpha_{4}^{{}^{\prime}}\\\
\beta_{1}^{{}^{\prime}}&\beta_{2}^{{}^{\prime}}&\beta_{3}^{{}^{\prime}}&\beta_{4}^{{}^{\prime}}\end{array}\right)$

be MSC of $\mathbf{A}$ with respect to $\mathbf{e}$. Then (4.1) in terms of
$A$ becomes

(4.2)
$\left\\{\begin{array}[]{rrr}\alpha_{1}^{{}^{\prime}}x_{1}^{2}+(\alpha_{2}^{{}^{\prime}}+\alpha_{3}^{{}^{\prime}})x_{1}x_{2}+\alpha_{4}^{{}^{\prime}}x_{2}^{2}-\lambda(x_{1},x_{2})x_{1}&=&0\\\
\beta_{1}^{{}^{\prime}}x_{1}^{2}+(\beta_{2}^{{}^{\prime}}+\beta_{3}^{{}^{\prime}})x_{1}x_{2}+\beta_{4}^{{}^{\prime}}x_{2}^{2}-\lambda(x_{1},x_{2})x_{2}&=&0\\\
\end{array}\right.$

We consider two options:

  1. (1)

Let $\mathrm{\lambda(x)}$ be not identically zero over
$\mathbb{F}^{2}\setminus\\{(0,0)\\}$. Then it is easy to see that
$x_{1}=0,x_{2}\neq 0$ ($x_{1}\neq 0,x_{2}=0$) implies
$\alpha_{4}^{{}^{\prime}}=0,\ \lambda(x_{1},0)=\alpha_{1}^{{}^{\prime}}x_{1}$
(respectively, $\beta_{1}^{{}^{\prime}}=0,\
\lambda(0,x_{2})=\beta_{4}^{{}^{\prime}}x_{2}$). Therefore, if $x_{1}\neq 0,\
x_{2}\neq 0$ then the system (4.2) can be rewritten as follows

$\left\\{\begin{array}[]{rrr}\alpha_{1}^{{}^{\prime}}x_{1}+(\alpha_{2}^{{}^{\prime}}+\alpha_{3}^{{}^{\prime}})x_{2}&=&\lambda(x_{1},x_{2})\\\
(\beta_{2}^{{}^{\prime}}+\beta_{3}^{{}^{\prime}})x_{1}+\beta_{4}^{{}^{\prime}}x_{2}&=&\lambda(x_{1},x_{2})\\\
\end{array}\right.$

In particular,
$(\alpha_{1}^{{}^{\prime}}-\beta_{2}^{{}^{\prime}}-\beta_{3}^{{}^{\prime}})x_{1}+(\alpha_{2}^{{}^{\prime}}+\alpha_{3}^{{}^{\prime}}-\beta_{4}^{{}^{\prime}})x_{2}=0$
and, if $\mathrm{Card}(\mathbb{F})>2$ (that is $\mathbb{F}\neq\mathbb{Z}_{2}$)
then the last system is noting but
$\alpha_{1}^{{}^{\prime}}=\beta_{2}^{{}^{\prime}}+\beta_{3}^{{}^{\prime}},\
\beta_{4}^{{}^{\prime}}=\alpha_{2}^{{}^{\prime}}+\alpha_{3}^{{}^{\prime}}$.
So, if $\mathbb{F}\neq\mathbb{Z}_{2}$ then
$\alpha_{1}^{{}^{\prime}}x_{1}+\beta_{4}^{{}^{\prime}}x_{2}=\lambda(x_{1},x_{2}),$
at least one of $\alpha_{1}^{{}^{\prime}},\beta_{4}^{{}^{\prime}}$ is not zero
and

$A=\left(\begin{array}[]{cccc}\alpha_{1}^{{}^{\prime}}&\alpha_{2}^{{}^{\prime}}&\beta_{4}^{{}^{\prime}}-\alpha_{2}^{{}^{\prime}}&0\\\
0&\beta_{2}^{{}^{\prime}}&\alpha_{1}^{{}^{\prime}}-\beta_{2}^{{}^{\prime}}&\beta_{4}^{{}^{\prime}}\end{array}\right)$

If $\mathbb{F}=\mathbb{Z}_{2}$ then
$\alpha_{1}^{{}^{\prime}}x_{1}+(a-\beta_{2}^{{}^{\prime}}-\beta_{3}^{{}^{\prime}})x_{2}=\lambda(x_{1},x_{2}),$
at least one of
$\alpha_{1}^{{}^{\prime}},a-\beta_{2}^{{}^{\prime}}-\beta_{3}^{{}^{\prime}}$
is not zero and one has

$A=\left(\begin{array}[]{cccc}\alpha_{1}^{{}^{\prime}}&\alpha_{2}^{{}^{\prime}}&a-\alpha_{1}^{{}^{\prime}}-\alpha_{2}^{{}^{\prime}}&0\\\
0&\beta_{2}^{{}^{\prime}}&\beta_{3}^{{}^{\prime}}&a-\beta_{2}^{{}^{\prime}}-\beta_{3}^{{}^{\prime}}\end{array}\right)$

  2. (2)

Let $\mathrm{\lambda(x)}$ be identically zero over
$\mathbb{F}^{2}\setminus\\{(0,0)\\}$. In this case the system (4.2) is
equivalent to
$\alpha_{1}^{{}^{\prime}}=\alpha_{4}^{{}^{\prime}}=\beta_{1}^{{}^{\prime}}=\beta_{4}^{{}^{\prime}}=\alpha_{2}^{{}^{\prime}}+\alpha_{3}^{{}^{\prime}}=\beta_{2}^{{}^{\prime}}+\beta_{3}^{{}^{\prime}}=0$
and therefore, we have

$A=\left(\begin{array}[]{cccc}0&\alpha_{2}^{{}^{\prime}}&-\alpha_{2}^{{}^{\prime}}&0\\\
0&\beta_{2}^{{}^{\prime}}&-\beta_{2}^{{}^{\prime}}&0\end{array}\right)$

These type of algebras were said to be zeropotent (see [9, 16, 17]).

Since now we know for a two-dimensional algebra given by MSC $A$ to be a
curled algebra, we can treat the description of two-dimensional curled
algebras by using Theorems 1, 2 and 3. For example, in the case of
$\mathbb{F}$ with $Char(\mathbb{F})\neq 2,3,$ a two-dimensional algebra given
by

$A=A_{1}(\mathrm{c})=\begin{pmatrix}\alpha_{1}&\alpha_{2}&1+\alpha_{2}&\alpha_{4}\\\
\beta_{1}&-\alpha_{1}&1-\alpha_{1}&-\alpha_{2}\end{pmatrix},$

where
$\mathrm{c}=(\alpha_{1},\alpha_{2},\alpha_{4},\beta_{1})\in\mathbb{F}^{4},$ is
a curled algebra if and only if $A$ has the following form

$A=\left(\begin{array}[]{cccc}\alpha_{1}^{{}^{\prime}}&\alpha_{2}^{{}^{\prime}}&\beta_{4}^{{}^{\prime}}-\alpha_{2}^{{}^{\prime}}&0\\\
0&\beta_{2}^{{}^{\prime}}&\alpha_{1}^{{}^{\prime}}-\beta_{2}^{{}^{\prime}}&\beta_{4}^{{}^{\prime}}\end{array}\right),$

that is

$\left\\{\begin{array}[]{cc}\alpha_{1}^{{}^{\prime}}=\alpha_{1}\\\
\alpha_{2}^{{}^{\prime}}=\alpha_{2}\\\
\beta_{4}^{{}^{\prime}}-\alpha_{2}^{{}^{\prime}}=1+\alpha_{2}\\\
0=\alpha_{4}\\\ 0=\beta_{1}\\\ \beta_{2}^{{}^{\prime}}=-\alpha_{1}\\\
\alpha_{1}^{{}^{\prime}}-\beta_{2}^{{}^{\prime}}=1-\alpha_{1}\\\
\beta_{4}^{{}^{\prime}}=-\alpha_{2}\par\end{array}\right.$

which happens if and only if
$\alpha_{1}=\frac{1}{3},\alpha_{2}=-\frac{1}{3},\alpha_{4}=0,\beta_{1}=0$.
Therefore, among the algebras $A_{1}(\mathrm{c})$ only
$A_{1}(\frac{1}{3},-\frac{1}{3},0,0)$ is curled.

Going through $A_{2}(\mathrm{c})-A_{13}$ in this manner in Theorem 1 one comes
to the following classification result.

######  Theorem 7.

Any non-trivial $2$-dimensional curled algebra over a field $\mathbb{F}$
$(Char(\mathbb{F})\neq 2,3)$ is isomorphic to only one of the following,
listed by their matrices of structure constants, such algebras:

  * •

$A_{1}(\frac{1}{3},-\frac{1}{3},0,0)=\begin{pmatrix}\frac{1}{3}&-\frac{1}{3}&\frac{2}{3}&0\\\
0&-\frac{1}{3}&\frac{2}{3}&\frac{1}{3}\end{pmatrix},$

  * •

$A_{3}(\alpha_{1},0,2\alpha_{1}-1)=\begin{pmatrix}\alpha_{1}&0&0&0\\\
0&2\alpha_{1}-1&1-\alpha_{1}&0\end{pmatrix},$ where $\alpha_{1}\in\mathbb{F}$

  * •

$A_{7}(\frac{1}{3},0)=\begin{pmatrix}\frac{1}{3}&0&0&0\\\
0&\frac{2}{3}&-\frac{1}{3}&0\end{pmatrix}$.

Among the listed algebras only $A_{3}(0,0,-1)$ is zeropotent.

Similarly, in $Char(\mathbb{F})=2,3$ cases the following results hold true.

######  Theorem 8.

Any non-trivial $2$-dimensional curled algebra over a field
$\mathbb{F}\neq\mathbb{Z}_{2}$ $(Char(\mathbb{F})=2)$ is isomorphic to only
one of the following, listed by their matrices of structure constants, such
algebras:

  * •

$A_{1,2}(1,1,0,0)=\begin{pmatrix}1&1&0&0\\\ 0&1&0&1\end{pmatrix}$,

  * •

$A_{3,2}(\alpha_{1},0,1)=\begin{pmatrix}\alpha_{1}&0&0&0\\\
0&1&1+\alpha_{1}&0\end{pmatrix},$ where $\alpha_{1}\in\mathbb{F}$

  * •

$A_{6,2}(1,0)=\begin{pmatrix}1&0&0&0\\\ 0&0&1&0\end{pmatrix}$

and among them only $A_{3,2}(0,0,1)$ is zeropotent.

######  Theorem 9.

Any non-trivial $2$-dimensional curled algebra over a field
$\mathbb{F}=\mathbb{Z}_{2},$ is isomorphic to only one of the following
listed, by their matrices of structure constants, such algebras:

  * •

$A_{1,2}(1,1,0,0)=\begin{pmatrix}1&1&0&0\\\ 0&1&0&1\end{pmatrix},$

  * •

$A_{3,2}(\alpha_{1},0,1)=\begin{pmatrix}\alpha_{1}&0&0&0\\\
0&1&1+\alpha_{1}&0\end{pmatrix},$ where $\alpha_{1}\in\mathbb{Z}_{2}$

  * •

$A_{4,2}(\alpha_{1},0,0)=\begin{pmatrix}\alpha_{1}&1&1&0\\\
0&0&1+\alpha_{1}&1\end{pmatrix},$ where $\alpha_{1}\in\mathbb{Z}_{2}$

  * •

$A_{6,2}(1,0)=\begin{pmatrix}1&0&0&0\\\ 0&0&1&0\end{pmatrix},$

  * •

$A_{7,2}(0,0)=\begin{pmatrix}0&1&1&0\\\ 0&1&0&1\end{pmatrix},$

  * •

$A_{10,2}(0)=\begin{pmatrix}1&1&1&0\\\ 0&1&1&1\end{pmatrix}$.

Among these algebras only $A_{3,2}(0,0,1)$ is zeropotent.

######  Theorem 10.

Any non-trivial $2$-dimensional curled algebra over a field $\mathbb{F}$
$(Char(\mathbb{F})=3)$ is isomorphic to only one of the following listed, by
their matrices of structure constants, such algebras:

  * •

$A_{3,3}(\alpha_{1},0,2+2\alpha_{1})=\begin{pmatrix}\alpha_{1}&0&0&0\\\
0&2+2\alpha_{1}&1+2\alpha_{1}&0\end{pmatrix},\ \mbox{where}\ \
\alpha_{1}\in\mathbb{F},$

  * •

$A_{4,3}(0,2)=\begin{pmatrix}0&1&1&0\\\ 0&2&1&2\end{pmatrix},$

  * •

$A_{11,3}(0)=\begin{pmatrix}0&1&1&0\\\ 0&0&0&2\\\ \end{pmatrix},$

Among the listed algebras only $A_{3,3}(0,0,2)$ is zeropotent.

##  5\. Two-dimensional endo-commutative curled algebras

In this section we make use the results of the last two sections to get a
classification of two-dimensional endo-commutative curled algebras up to
isomorphism.

######  Theorem 11.

Any non-trivial two-dimensional endo-commutative curled algebra over a field
$\mathbb{F}$ $(Char(\mathbb{F})\neq 2,3)$ is isomorphic to only one of the
following, listed by their matrices of structure constants, such algebras:

  * •

$A_{3}(\alpha_{1},0,2\alpha_{1}-1)=\begin{pmatrix}\alpha_{1}&0&0&0\\\
0&2\alpha_{1}-1&1-\alpha_{1}&0\end{pmatrix},$ where $\alpha_{1}\in\mathbb{F},$

  * •

$A_{7}(\frac{1}{3},0)=\begin{pmatrix}\frac{1}{3}&0&0&0\\\
0&\frac{2}{3}&-\frac{1}{3}&0\end{pmatrix}$

and among them only $A_{3}(0,0,-1)$ is zeropotent.

######  Theorem 12.

Any non-trivial $2$-dimensional Endo-commutative curled algebra over a field
$\mathbb{F}\neq\mathbb{Z}_{2}$ $(Char(\mathbb{F})=2)$ is isomorphic to only
one of the following, listed by their matrices of structure constants, such
algebras:

  * •

$A_{3,2}(\alpha_{1},0,1)=\begin{pmatrix}\alpha_{1}&0&0&0\\\
0&1&1+\alpha_{1}&0\end{pmatrix},$ where $\alpha_{1}\in\mathbb{F},$

  * •

$A_{6,2}(1,0)=\begin{pmatrix}1&0&0&0\\\ 0&0&1&0\end{pmatrix}$.

Among the algebras listed above only $A_{3,2}(0,0,1)$ is a zeropotent algebra.

.

######  Theorem 13.

Any non-trivial two-dimensional endo-commutative curled algebra over a field
$\mathbb{F}=\mathbb{Z}_{2}$ is isomorphic to only one of the following listed,
by their matrices of structure constants, such algebras:

  * •

$A_{3,2}(0,0,1)=\begin{pmatrix}0&0&0&0\\\ 0&1&1&0\end{pmatrix},$

  * •

$A_{3,2}(1,0,1)=\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\end{pmatrix},$

  * •

$A_{4,2}(0,1,0)=\begin{pmatrix}0&1&1&0\\\ 0&0&1&1\end{pmatrix},$

  * •

$A_{4,2}(1,1,0)=\begin{pmatrix}1&1&1&0\\\ 0&0&0&1\end{pmatrix},$

  * •

$A_{6,2}(1,0)=\begin{pmatrix}1&0&0&0\\\ 0&0&1&0\end{pmatrix},$

  * •

$A_{7,2}(0,0)=\begin{pmatrix}0&1&1&0\\\ 0&1&0&1\end{pmatrix},$

  * •

$A_{10,2}(0)=\begin{pmatrix}1&1&1&0\\\ 0&1&1&1\end{pmatrix}.$

Here, only $A_{3,2}(0,0,1)$ is zeropotent.

Note that the endo-commutative curled algebras given in [26] can be found in
Theorem 13 as $A_{3,2}(0,0,1)\simeq C_{4},\ A_{3,2}(1,0,1)\simeq C_{3},\
A_{4,2}(0,1,0)\simeq C_{12},\ A_{4,2}(1,1,0)\simeq C_{7},\ A_{6,2}(1,0)\simeq
C_{2},\ A_{7,2}(0,0)\simeq C_{13}$ and $A_{10,2}(0)\simeq C_{1}.$

######  Theorem 14.

Any non-trivial two-dimensional endo-commutative curled algebra over a field
$\mathbb{F}$ $(Char(\mathbb{F})=3)$ is isomorphic to only one of the following
listed, by their matrices of structure constants, such algebras:

  * •

$A_{3,3}(\alpha_{1},0,2\alpha_{1}-1)=\begin{pmatrix}\alpha_{1}&0&0&0\\\
0&2\alpha_{1}-1&1-\alpha_{1}&0\end{pmatrix},$ where $\alpha_{1}\in\mathbb{F},$

  * •

$A_{11,3}(0)=\begin{pmatrix}0&1&1&0\\\ 0&0&0&2\end{pmatrix},$

Among the algebras listed above only $A_{3,3}(0,0,-1)$ is zeropotent.

##  6\. Acknowledgement

We would like to thank Professor K.Shirayanagi for pointing out the authors
attention to an algebra that was missed in Corollary 3.1.

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