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\title[ algebraic groups and triangular quasi-quadratic systems]{ algebraic groups and triangular quasi-quadratic systems}
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\author[]{A. Molkhasi }

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\authorsaddresses{Department of Mathematics, Faculty of Mathematical Sciences,\\
University of Farhangian-Iran\\
molkhasi@gmail.com\\
}
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\subjclass[2010]{Primary: 20F10; Secondary: 20D60.}

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\keywords{Algebraic set, Equationally noetherian groups, Triangular
quasi-quadratic systems, strongly algebraically closed}



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\begin{abstract}
We introduce the notion of  $q^\prime$-compactness for connected algebraic groups. We characterize the conditions when the lattice of all closed connected subgroups of connected algebraic group  turn out to be prevalent  in determining  the algebraic structure of the group is a strongly algebraically closed algebra. Also, we study the relationships between  the families of groups  and the triangular quasi-quadratic systems.
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\section{Introduction}
The objective of this paper is to study the algebraically closed objects in some classes of algebras. An algebraically closed algebraic structure is formulated and recent studies in algebraic  geometry  by   A. Myasnikov, E. Daniyarova, O. Kharalampovich, V. Remeslennikov and others (see  \cite{Romankov2}, \cite{Dani1}-\cite{Dani44}, \cite{Myasn998}, \cite{Myasn98}, \cite{Modabberi}, and \cite{Shahryari})  are further studied and discussed. We observe that  R. Schmid  \cite{Sch}  has first considered the  subgroup lattices of a group and  A. Shevlyikov has proposed  some new  problems in universal  geometry which are  related  by  Boolean  equations in \cite{Shevl}.  In this aspect,   A.  Shevlyikov  and A.  Molkhasi in \cite{molkhasi2018}  further proved  that if a complete Boolean lattice  is  $q^\prime$-compact,  then  it is a  strong algebraically closed lattice.\\
It is noted that G. Gratzer  \cite{G68}  observed that all complemented  distributive lattices  are closely related to  Boolean algebras and A. Sherlyikov \cite{Shevl} also observed that  two Boolean algebras $B_1,\, B_2$  are geometrically  equivalent if and only if  any system equations  over  $B_1,\, B_2$ are isomorphic.\\
In this paper, we first define  the concept  of $q^\prime$-compactness for a connected algebraic group G so that we obtain some necessary conditions  for those   lattices of   its closed connected subgroup G   to be a strong closed algebraic closed algebras.  Finally,  we  also study the  the triangular  quasi- quadratic systems on free orderable groups. Furthermore, we also consider  the kernel of  Frobeniu  groups of even order,  the equally Noetherian  groups to torsion-free  hyperbolic  groups and  Engel groups \cite{Zanr}.



\section{ strongly algebraically closed  connected algebraic groups }

In this section we mention some basic notions of systems of equations and algebraic sets. We refer the reader to \cite{Romankov9} for details. \\

Throughout this section,   $\mathcal{L}$ is an algebraic language and $A$ is  an algebra of type  $\mathcal{L}$.
 If we attach the elements of $A$ as constants to $\mathcal{L}$, then  we simply  denote   the new language by  $\mathcal{L}(A).$  Now,  we call  an algebra  $B$
of type $\mathcal{L}(A)$ an  $A-$algebra  of the map $a\hookrightarrow a^B$  is an  embedding of $A$ into $B$. Note that $a^B$ denotes the interpretation of the
constant symbol $a$ in $B$. We assume that $X = \{x_1,\,\ldots,\,x_k\}$ is a finite set of variables. We
denote the term algebra in the language $\mathcal{L}$ and variables from $X$ by
$T_\mathcal{L}(X)$, and similarly the term algebra in the extended language $\mathcal{L}(A)$
will denoted by $T_{\mathcal{L}(A)}(X)$. An equation is a pair $(p, \,q)$ of the elements of the term algebra $T_\mathcal{L}(X)$.
In many cases, we assume that such an equation is the same as the atomic formula $p(x_1,\,\ldots,\,x_n) \approx q(x_1,\,\ldots,\,x_n)$ or briefly $p \approx q$.
Hence, the set $At_\mathcal{L}(X)$ of atomic formula in the language $\mathcal{L}$. Any subset $S \subseteq�At_\mathcal{L}(X)$ is called a \textit{system of
equations} in the language $\mathcal{L}$. Suppose $B$ is an $A$-algebra. An element $(b_1,\, \ldots ,\, b_n)\in B^n$ will be denoted by $\overline{b}$. Let $S$ be a system of equations with coefficients in $A$. Then the set
$$V_B(S) = \{\overline{b}\in B^n \,\,|\,\, �\forall (p \approx q)\in S,\, p^B(\overline{b}) = q^B(\overline{b})\}$$
is called an \textit{algebraic set}. A structure $A$ is algebraically closed iff every finite set of equations with parameters in $A$   has a solution in some extension   of $A$.

We recall from \cite{G68} that a lattice L is relatively complemented if every interval
$[x,\, y]$ of lattice is complemented. A complement in $[x,\, y]$ of $a\in [x,\, y]$ is called a relative complement of $a$. We say that the algebra $A$ is finitary equational Noetherian, if every finitary system of equations in the language $\mathcal{L}(A)$ is reducible over $A$ to a finite system.

Let $G$ is connected algebraic group. Then the set  $\Lambda G$ of an algebraic  group    $G$ forms a lattice with the definition $A\wedge B=A\cap B $  and $A\vee B $ be the algebraic subgroup generated by $A$ and $B$.
\begin{theorem}
Suppose $G$ is connected algebraic group and  $\Lambda G$ is the complete lattice   of all closed connected subgroups of $G$. If $G$ is $IM$-group and $\Lambda G$ is $q^\prime$-compact, then  $\Lambda G$ is  a strongly algebraically closed algebra.
\end{theorem}

\begin{proof}
By Theorem B of \cite{P},  $\Lambda G$ is a relatively complemented lattice iff  $G$ is $IM$-group. Thus, $\Lambda G$ is a relatively complemented lattice. On the other hand, we know that $\Lambda G$ is algebraically closed lattice iff $\Lambda G$ is a relatively complemented lattice \cite{Sch}. Notice that $\Lambda G$ is a complete lattice. Therefore, by \cite{Shevl}, $\Lambda G$ is a weak equational Noetherian Boolean algebra and a finite system $T$  equivalent to $S$ over $\Lambda G$. Using the result \cite{Sch} of Schmid, every finite $S_0\subseteq S$ is consistent and has a solution in $\Lambda G$ and then $S$ has a solution in some ultra-power $B={\Lambda G}^I/\mathcal{U}$. Note that $B$ is also a distributive
lattice, and since it is an elementary extension of $\Lambda G$, and $\Lambda G$ is $q^{\prime}$-compact then $Rad_{\Lambda G}(S)=Rad_B(S).$ On the other hand, we have $
Rad_{\Lambda G}(S)=Rad_{\Lambda G}(T).$ Since $T$ is finite, we have $Rad_{\Lambda G}(T)=Rad_B(T).$ This shows that $S$ and $T$ are equivalent over $B$. Therefore, $T$ has a solution in $B$ and consequently in ${\Lambda G}$. Thus $S$ has a solution in ${\Lambda G}$.
Note that  $T$ is a finite system in the language $\mathcal{L}({\Lambda G})$. But by introducing a finite number of new
variables and a finite number of new equations we can transform it to a finite system in $\mathcal{L}({\Lambda G})$. To do this, we perform the following actions:\\

1- If $T$ contains the boolean constants $0$ and $1$, then there will be no change, since $0, 1\in {\Lambda G}$. \\

2- If $T$ contains therm $x^{\prime}$, then we introduce a new variable $y$ and insert new equations $x\wedge y\approx 0$ and $x\vee y\approx 1$, instead.\\

3- If there appears a term of the form $a^{\prime}$, then again there will not be any changes.
\end{proof}
\section{Systems of equations and the NTQ systems }
In this section, we recall some definitions that we need to them. Suppose that $G$ is a fixed group and $X = \{x_1,\,\ldots,\,x_k\}$ is a finite set of variables. The free product of $G$ and the free group $F(X)=\langle x_1,\ldots,\,x_k \rangle$ freely generated by $\{x_,\ldots,x_k\}$ denoted by:\\
$$G[X] =G[x_1,\ldots,\,x_k] =G\ast F(X).$$
We sometimes say that $\{x_1,\ldots,x_k\}$ freely generates the free $G$-group $G[X]$. Recall that $G$-groups can be linked to algebras over a unitary commutative ring, more specially a field, with $G$ playing the role of the coefficient a ring.
We  now call  a group $H$ is a $G$-group if  it contains a  designed copy of $G$ which  will for the most part identify with $G$. Obviously,  the group  $G$ itself is a  $G$-group. Let  $H$ be a  $G$-group. Then we write $H^k= \{(a_1,\ldots,\,a_k)\,\,|\,\,a_i\in H\}$. We can easily see that  $H^k$  is  an affine $k-$space over $H$. The elements  of $H^k$ can be regarded as points. A subset  of $G^n$ is closed if it is the intersection of an arbitrary number of
finite unions of algebraic sets. Thus, we  have defined a topology on  $G^n$ which is  the well known Zariski  Topology on  $G^n$. Now,  we call   a group  $G$  equationally  Noetherian  if and only if  for each  $n\geq 1$, the Zariski   Topology  on  $G^n$ is Noetherian. With the above notion,  we call a  subgroup  $F$ of the group  $G$  Noetherian  if for every  $k$ greater than $0$  and every subset  $S$ of $G[x_1,\ldots,\,x_k]$ , there exists  a finite subset  $S_0$ of $S$ such that  $V_H(S)=V_H(S_0),$ as usual. Now, we observe that  $ V_H(S) = \{v\in H^n\,|\,\,f(v)=1, \, \textrm{for all}\,\, f\in S\}$. When $G=H$, we simply call the group $G$ equational Noetherian.
Recall that term
$$
v=(a_1,\ldots,\,a_k)\in H^k
$$
is a root of a polynomial in the non-commuting variables $x_1,\ldots,\,x_k$, with coefficient
in $G$ if $f(v)= f(a_1,\ldots,\,a_k,\,g_1,\ldots,\,g_t)=1$, where the constants $g_1,\ldots,\,g_t \in G $ \cite{Shevl}.\\

We continue to study  the systems of equations over free product of groups which were considered by M. Casals-Ruiz and I.V. Kazachkov in \cite{Kazachkov}. For an  arbitrary element  $s\in G[x]$ the  for all equation  $S=1$ can be  treated  as an equation over $G$, where   $G$ is a fixed group generated   by a set $A$  and $X = \{X_1,\, X_2,\, \ldots,\, X_n\}$. In general, for a subset $S \in G[X]$, the formal   equation  $ S=1$ can be regarded as a system of equations over $G$  with coefficients in $A$.

A finite system of equations $S = 1$ is called \textit{triangular quasi-quadratic} over group $G$ if it can be partitioned
into the following subsystems\\
$$
S_1(X_1,\, X_2,\, \ldots,\, X_n,\, \textit{A}) = 1,
$$
$$
S_2(X_2,\, \ldots,\, X_n,\, \textit{A}) = 1,
$$
$$
\vdots
$$
$$
S_n(X_n,\, \textit{A}) = 1,
$$

where for each $1\leq i \leq n$  one of the following holds:\\

(1) $S_i$ is a quadratic equation in variables $X_i$;\\
(2) $S_i = \{[y, z] = 1,\, [y, u] = 1,\, [z, u] = 1 \,|\, y,\, z \in X_i\}$, where $u$ is a group word in $X_{i+1} \cup\ldots\cup X_n\cup A$;\\
(3) $S_i = \{[y,\, z] = 1 \,|\, y, \,z \in X_i\}$;\\
(4) $S_i$ is the empty equation.\\

With the above notations, we define $G_i = G_{R(S_i,\,\ldots,\,S_n)}$ and  we set  $G_{n+1}=G.$ Now, we  call the triangular quasi-Quadratic system $S=1$, the non-degenerated  or simply  NTQ if the following two conditions hold:\\

(A) Each system $S_i = 1$, where $X_{i+1},\,\ldots,\, X_n$ are viewed as the corresponding constants from $G_{i+1}$
(under the canonical maps $X_ j\longrightarrow G_{i+1},\, j = i +1,\,\ldots,\,n)$ has a solution in $G_{i+1}$;\\

(B) The element in $G_{i+1}$ represented by the word $u$ from case (3) above, is not a proper power in $G_{i+1}$.

 It is noted that the Frobenius groups and  the $2$-Frobenius  groups were  first studied by  G.Y. Chen  in \cite{Chen95}.  Throughout this section,  we   let $G$  be a  finite group acting transitively  on a set  $X$ , with  more than one elements. It is  known that  for an element $g\in G$ and two distinct elements  $x$ and $y$ if $gx=x$ and $gy=y$, then  $g=1$ (see \cite{Chen95}). Now,  we state the  following main theorem in our paper.
\begin{theorem}
 Let $G$   be a the kernel of a  Frobenius group of even order. If there exists an abelian group such that the its subgroups lattice is isomorphic to the lattice of normal subgroups of $G$, then $G$ is an equationally Noetherian group and every closed set in affine $k$-space $H^k$ is a union of finitely many images of algebraic sets of the NTQ systems, which $H=G[x_1,\ldots,\,x_k]$ is the free product of $G$ and the free group $F(X)=\langle x_1,\ldots,\,x_n \rangle$ freely generated by $\{x_1,\ldots,x_k\}$.
\end{theorem}

\begin{proof}
Let $G$ be the kernel of a Frobenius group of even order. By (\cite{Iranmanes}, Lemma 2.1), $G$ is a nilpotent group.
Suppose that there exists an abelian group $C$ such that $N(G)$ the lattice of normal subgroups of $G$ is isomorphic to $N(C)$, the subgroup lattice of $C$, and
$$ \theta : N(G)\rightarrow N(C)$$
be an isomorphism and $G$ is non-abelian group. We observe that the group $C$ is decomposed as a direct sum of its $p$-components such that $C=\bigoplus_p C_p$ and we will have $G= \prod_p\theta^{-1}(C_p)$. Since $G$ is non-abelian, so there exists a prime $q$ such that $\theta^{-1}(C_q)$ is a non-abelian because $G$ is a torsian group (see \cite{Sch}).
Consequently, we  may  assume that  $C$ is a  $G$-group. By apply  Theorem  9.1.11,  in \cite{Sch}, we can easily see that   the group $C$ is a locally  cyclic group and the lattice of groups of $C$ forms a chain (see [23]). Since  $G$ is not  abelian, then the nilpotency class $t$ of $G$  is greater than  $2$ for $t\in \mathbb{N}$ and  we can find a $k$ such  that  $G_{t-1}=M_k$, where  $N(G)=\{1< M_1< \ldots M_k< \ldots\}$ (see \cite{Sch}). Hence,  we can immediately see that  $G[X]=G[x_1,....,x_k]$ is a  $G-$equationally  Noetherian  group. Now, we consider group $G/ G_{t-2}$. We set $H=G/ G_{t-2}$ and $Z(H)=G_{t-1}/ G_{t-2}$. It is clear that  $H/ Z(H)$ is an abelian group. If we note that the form subgroups of $H/ Z(H)$ and using Section 3 of \cite{Fuchs} we can say that $H/ Z(H)$ is union of an ascending chain of cyclic subgroups. We know that if $x,\,y\in H$, then we can find an element $u$ of $H$ such that $xZ(H),\,yZ(H) \in \langle uZ(H)\rangle$ and $x =u^kv$, $y=u^\ell z$ for some integers $k$ and $\ell$ and some elements $v,\,z\in Z(H)$. Then $xy=yx$ and $H$ is an abelian that it is a contradiction. So, $G$ is an abelian. On the other hand, every abelian group is equationally Noetherian (see Theorem 1 of \cite{Romankov9}). Also, we know if $G$ is an equationally Noetherian group, then free $G$-group $G[x_1,\ldots,\,x_k]$ is $G$-equationally Noetherian (see \cite{Romankov9}).  So, $G[X]=G[x_1,\ldots,\,x_k]$ is $G$-equationally Noetherian group. By
(\cite{Romankov9}, Corollary 11),  we have that if $H$ is $G$-group and it is a equationally Noetherian group, then every closed set in $H^k= \{(a_1,\ldots,\,a_k)\,\,|\,\,a_i\in H\}$, affine $k$-space over $H$, is a finite union of algebra sets. Here, we set $H=G[X]$. By \cite{Kazachkov}, every algebraic set over $G$ can be decomposed as a union of finitely many images of algebraic sets of the NTQ systems. Finally, we conclude that every closed set in affine $k$-space $H^k$ is union of finitely many images of algebraic sets of the NTQ systems.
Thus,  we have proved that   if $G$ is the kernel  of a  Frobenius group  of even order and the lattice of normal subgroups of $G$ is isomorphic to the subgroup lattice of an abelian group,  then  every close set in an affine $k$-space over $H^k$ is union of finitely many images of algebraic sets of the NTQ systems.
\end{proof}

\begin{corollary}
Let $G$ be the kernel of a Frobenius group of even order and $N$ be a normal subgroup of $G$ which is a finite union
of algebraic sets over $G$.  If there exists an abelian group such that the its subgroups lattice is isomorphic to the lattice of normal subrgoups of $G$, then $G/N$ is also an equationally Noetherian.
\end{corollary}

\begin{proof}
By Theorem 4.1,  $G$ is an abelian group. In \cite{Romankov9}, it is proved that abelian groups
are equationally Noetherian. By (\cite{Romankov2}, Theorem 1), we see that  of $G$  is equationally  Noetherian  and $V$ is a  normal  subgroup of  $G$ which  is a finite  number of alphabets  over $G$, then  $G/N$ is also  equationally  Noetherian. This completes the proof.
\end{proof}

We recall the definition of a $CSA$-group with an involution. Note that $CSA$-groups are in particular commutative transitive, i.e. groups in which the
relation $[x,\, y]=1$ is an equivalence relation on the set of nontrivial elements, or equivalently in which the centralizer of every nontrivial element is an abelian. For more details about $CSA$-groups, we refer the reader \cite{Gi} and \cite{My}. A \textit{$CSA$-group} is a group  $G$ in which every maximal abelian subgroup $\textbf{A}$ is malnormal i.e. satisfies $\textbf{A} \cap \textbf{A}^g = {1}$ for all
$g\in G\backslash \textbf{A}$. \\

A presentation$(\verb"A",  \mathfrak{R})$ is called finite if both sets $\verb"A"$ and $\mathfrak{R}$ are finite. A group is \textit{finitely presented} if it has a finite presentation. Let $\lambda$ be a number in $[0, 1]$. Let $(\verb"A",  \mathfrak{R})$ be a group presentation,
and $\widehat{\mathfrak{R}}$ be the symmetrization of $\mathfrak{R}$. Then the set $\mathfrak{R}$, or the
presentation $(\verb"A",  \mathfrak{R})$, is said to satisfy the condition $C^\prime (\lambda)$ if $|X| \leq \lambda |\verb"R"|$ for
any $\verb"R"\in \widehat{\mathfrak{R}}$ and for any subword $X$ of $\verb"R"$ which is a piece relative to $\widehat{\mathfrak{R}}$.\\

We say that a group presentation $(\verb"A",  \mathfrak{R})$ is \textit{singularly aspherical} if it is diagrammatically aspherical, concise,
and no element of $\mathfrak{R}$ can be decomposed as a concatenation of several copies.


\begin{theorem}
Suppose $G$ is a finitely presented group with an involution which has a presentation satisfying $C^\prime(\frac{1}{6})$. If $G$ is a group with singularly spherical presentation. Then every closed set in affine $k$-space $H^k$ is a union of finitely many
images of algebraic sets of the NTQ systems, which $H=G[x_1,\ldots,\,x_k]$ is the free product of $G$ and the free group $F(X)=\langle x_1,\ldots,\,x_n \rangle$ freely generated by $\{x_1,\ldots,x_k\}$.
\end{theorem}

\begin{proof}
By (\cite{90}, Theorem 33), a finitely presented group which has a presentation satisfying $C^\prime(\frac{1}{6})$ is hyperbolic. On the other hand, we know every group with singularly spherical presentation is torsion-free (\cite{mor}, Lemma 64). Thus, $G$ torsion-free hyperbolic group. By (\cite{My}, Proposition 12), $G$ is a $CAS$-group. So by (\cite{My}, Remark 7), $G$ is an abelian group. Therefore, the assertion holds by Theorem 4.1.
\end{proof}

We recall the definition of an $H$-limit group. Suppose $H$ and $G$ are groups. Then
$(f_i\,|\,f_i : G \longrightarrow H,\, i\in \mathbb{N})$ is a sequence of morphisms from $G$ to $H$. We say that $f_{i\in \mathbb{N}}$
is convergent or stable if for every $g \in G$ one of the following sets is finite:\\
$$\{i\in \mathbb{N}\,|\,f_i(g)=1\},\,\,\{i\in \mathbb{N}\,|\,f_i(g)\neq1\}.$$

We set $\underrightarrow{\ker}(f_i)=\{g\in G\,|\,|\{i\in \mathbb{N}\,|\,f_i(g)\neq1\}| \,\textrm{is finite}\}.$\\


A group $K$ is said to be \textit{$H$-limit} if there exist a group $G$ and a converging sequence
$(f_i\,|\,f_i : G \longrightarrow H,\,\, i\in \mathbb{N} )$ such that $k=G/\underrightarrow{\ker}(f_i)$, where $(f_i\,|\,f_i : G \longrightarrow H\,\, i\in \mathbb{N})$ is a sequence of morphisms from group $G$ to group $H$.\\


Recall that limit groups are heavily connected to residual properties. Let $G$ be a group
and $\mathcal{K}$ a class of groups. We say that $G$ is residually-$\mathcal{K}$ (or that $\mathcal{K}$ separates
$G$) if for every $g \in G$  there exist $\textbf{K}\in \mathcal{K}$ and a morphism $f : G \longrightarrow \textbf{K}$
such that $f(g) \neq 1$.
\begin{corollary}
Let $G$ be a $CSA$-group with an involution. Then any finitely generated $G$-limit group is an equationally Noetherian group.
\end{corollary}

\begin{proof}
Since, $G$ is a $CSA$-group with an involution, then  $G$ is an equationally Noetherian group. Now, we prove that any
finitely generated $G$-limit group is equationally Noetherian. We have $G$ is finitely generated $G$-pseudo-limit group. Then any sequence of
epimorphisms of finitely generated residually-$G$ groups is also a sequence of epimorphisms of finitely generated $G$-pseudo-limit groups; thus terminates after
finitely many steps $G$ is an equationally Noetherian. So, $G$ is an equationally Noetherian group (\cite{Hou}, Corollary 2.10).
\end{proof}
\section{The NTQ systems for Engel groups }

In this section we recall Engel groups of Lie algebras. We refer the reader to \cite{BLY}, \cite{Has}, and \cite{Shumy} for details. Engel groups and left and right Engel elements in groups have been extensively
studied since the 1950�s and even justify a subsection in the AMS classification scheme. An element $x$ of a group $G$ is said a (left) \textit{Engel element} if
for any $g \in G$ there exists $n=n(x,\,g)\geq 1$ such that $[g_n,\, x]=1$. Suppose first that $[g_0,\, x]=g$ and the
commutator $[g_n,\, x]=1$ is defined recursively by the rule $$[g_n,\, x]=1=[[g_{n-1},\, x],\,x].$$
If $n$ can be chosen independent of $g$, then $x$ is a (left) $n$-Engel element. A group $G$ is called an \textit{$n$-Engel group} if all elements of
$G$ are $n$-Engel. Recently groups with $n$-Engel word-values were considered \cite{Bastos}, \cite{Shumyatsky}.\\

Recall that a partially ordered set $(L,\,\leq)$ is a lattice if $\sup\{a,\,b\}$ and $\inf\{a,\,b\}$ exist
for all $a,\,b\in L$. The group $G$ is called \textit{orderable} if there exists a full order relation $\leq$ on the set
$G$ such that every pair of elements has a least upper bound and greatest
lower bound and for any $x,\, y,\,a, \,b\in G$:
 $$x\leq y \Longrightarrow axb \leq ayb.$$
By $\gamma_i(G)$, we mean the $i$-th term of the lower central series of a group $G$. Now, we will require the following Theorem, due to Burns and Medvedev \cite{Bur}.

\begin{theorem}
Let $n\geq 1$. Then exist constant $c=c(n)$ and $e=e(n)$ depending only on $n$ such that, if $G$ is a
finite $n$-Engel group, then the exponent of $\gamma_c(G)$ divides $e$.
\end{theorem}
We are now ready to prove the following theorem.
\begin{theorem}
Let $m,\, n$ be positive integers and $G$ be orderable or free group. For all $x\in G$, if $x^m$ is $n$-Engel
and there exists an abelian group such that the lattice of normal subgroups of $G$ is isomorphism to the subgroup
lattice of it, then $G$ is equationally Noetherian group and every closed set in $H^k$ is a union of finitely many
images of algebraic sets of NTQ systems, which $H=G[x_1,\ldots,\,x_k]$.
\end{theorem}

\begin{proof}
In the free group $G$ with $\aleph$ generators, let $[G,\,G]=G_1$, and let $G_{t+1}=[G,\,G_t]$, where $[H,\,K]$
stands for the subgroup generated by all commutator $x^{-1}y^{-1}xy,\,x\in H,\,y\in K$. Then, by a theorem of
Magnus and Witt we will have $\bigcap G_t=0$, the group identity. On the other hand, we know that if a group $G$
has a well-ordered central series ending with the identity such that all quotient-groups $G_t\diagup G_{t+1}$ are
torsion-free, then $G$ is orderable. Using (\cite{Shumy}, Theorem 1.2), we observe that $G$ is nilpotent. Because
assume that $K$ is f.g torsion-free nilpotent. Then it is a residually finite $p$-group which $p$ is a prime number.
As a result for any finitely generated subgroup $K$ of $G$ we have $\gamma_c(H)=1$ and here $\gamma_c(G)=1$ and $G$ is
nilpotent of class at most $c-1$ which $c=c(n)$ in Theorem 3.2.  Notice that, we proved that $G$ is nilpotent.
By applying that (\cite{{Breaz}}, Theorem 2), $G$ is abelian group. On the other hand, every abelian group is
equationally Noetherian and by attention if $G$ is an equationally Noetherian group, then free $G$-group
$H=G[x_1,\ldots,\,x_k]$ is $G$-equationally Noetherian. Also, if $H$ is a $G$-equationally Noetherian group,
then every closed set in $H^k$ is a finite union of algebraic sets, see \cite{Romankov9}. As a result, it is a
union of finitely many images of algebraic sets of NTQ systems (see \cite{Kazachkov}).
\end{proof}

\begin{example}
Let $m,\, n$ be positive integers and $G$ be orderable or free group and $N$ be a subgroup of $G$ which is a finite union
of algebraic sets over G. For all $x\in G$, if $x^m$ is $n$-Engel and there exists an abelian group such that the lattice
of normal subgroups of $G$ is isomorphism to the subgroup lattice of it, then $G/N$ is also equationally Noetherian. In Theorem 6.2, we proved that $G$ is abelian group. Because in \cite{Romankov9}, it is proved that abelian groups are equationally Noetherian and consequently, $G/N$ is
equationally Noetherian (see \cite{Bu}).

\end{example}


\begin{example}
Let $m,\, n$ be positive integers and $G$ be orderable or free group. For all $x\in G$, if $x^m$ is $n$-Engel and there
exists an abelian group such that the lattice of normal subgroups of $G$ is isomorphism to the subgroup lattice of it,
then any finitely generated $G$-limit group is equationally Noetherian group. Because we proved that $G$ is abelian group and as a result, $G$ is equationally Noetherian group. Now, by applying that Corollary 2.10 of \cite{Hou}, completes the proof.
\end{example}

\textbf{Conclusion}\\
Another description of the solution sets of a  system of equations  was first given  by  M. Casals-Ruiz  and  I.V. Kazachkov \cite{Kazachkov}  in terms of the NTQ system ( $\omega$  residually free towers). We notice that  the NTQ  system has now become an essentially  tool  in the study if  the fully  residually  free  groups (limit groups). We conclude that exist the relationships between  the families of groups  and the triangular quasi-quadratic systems and every closed set in $H^k$ union of finitely many images of algebraic sets of NTQ systems, which $H$ is a $G$-equationally Noetherian group.\\

\textbf{Declarations}\\
Conflicts of interest: The author declare that there is no conflict of interests
regarding the publication of this paper.

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