Characteristic Sequence of Strongly Minimal Directed Single Graphs of 1-Arity

Abstract

In this paper, we will classify the strongly minimal directed single graphs of 1-arity by axiomatizing the theory of characteristic sequence of such a graph. Then we will show this theory is complete by using Łos-Vaught test. Complete theory is important to capture all the models of the theory and hence can be applied on mathematical structures which meet such a theory. The theory of algebraically closed fields with a given characteristic is complete. Thus, in this paper we will classify the strongly minimal directed single graphs of 1-arity with given characteristic sequence which can be applied on many mathematical structures not only algebraically closed fields.

1. Introduction and Preliminaries

The theory of algebraically closed fields with given characteristic is complete which means that we can capture all such fields that meat this theory. In this research we will first define directed single graphs of 1-arity and the construction of such graphs by defining the components of these graphs. Hence we can after present the theory of these graphs in general. This theory has many axioms which will be presented. However, not all these axioms can give a complete theory. Thus we will give a characteristic sequence which capture the strongly minimal directed single graphs of 1-arity, and algebraically close fields are strongly minimal [1,2] and show the theory of this sequence is uncountably categorical in all uncountable cardinals $\kappa$ ans hence is complete via Łos-Vaught test.

We introduce now preliminaries that are needed in this research, ref. [3] is the main source of these preliminaries and recommended for more details.

A sequence of parameters $(a_1,\dots ,a_n)$ in the graph G is written as $\overline{a}\in G$. We write $\phi (\overline{v},\overline{u})$ as formula in the tuples $\overline{v}$ and $\overline{u}$ of vertices. If the formula $\phi (\overline{a})$ is true in G then we write $G\vDash \phi (\overline{a})$ The cardinality of the graph G is denoted by $|G|$. Let $F\subseteq G$. Then $L_F$ is the language obtained by adding constant symbols to L for each element in F and $T{h}_F(G)$ is the set of all $L_F$-sentences true in G. The notation ${\mathbb{N}}^{+}$ stands for the set of positive integers.

Definition 1

(p. 117, [4]). Let $\mathcal{M}$ be an L-structure and $A\subseteq M$. Let $X\subseteq M^n$. We say X is definable with parameters from A if and only if there is an L-formula $\phi (x_1,\dots ,x_n,y_1,\dots ,y_m)$ and elements $a_1,\dots ,a_m\in A$ such that $X=\{(x_1,\dots ,x_n)\in M^n:\mathcal{M}\vDash \phi (x_1,\dots ,x_n,a_1,\dots ,a_m)\}$.

Theorem 1

([5]). The Łos-Vaught test state that a theory is complete if it is uncountably categorical for each uncountable cardinal κ and it has no finite model.

Definition 2

(p. 1, [6]). An infinite definable set $X\subseteq M^n$, where X is definable with parameters, is called minimal if every definable (with parameters) subset of X is either finite or cofinite. If $\phi (\overline{x},\overline{a})$ is the formula that defines X, then $\phi (\overline{x},\overline{a})$ is minimal. We say that X and $\phi (\overline{x},\overline{a})$ are strongly minimal if $\phi (\overline{x},\overline{a})$ is minimal in any elementary extension $\mathcal{N}$ of $\mathcal{M}$.

Theorem 2

(p. 211, [3]). Suppose T is a strongly minimal theory in a countable language. If $\kappa \ge {\aleph }_1$ and $\mathcal{M}$, $\mathcal{N}\vDash T$ with $|\mathcal{M}|=|\mathcal{N}|=\kappa$, then $\mathcal{M}\cong \mathcal{N}$.

Definition 3.

A graph G is a mathematical structure consists of the ordered pair $(V,E)$ where V is a set of vertices and E is a relation symbol defined by $E=\{\{u,v\}|u,v\in V^2\}$.

If G is directed graph, then $E=\{(u,v)|u,v\in V^2\}$ where the edge starts from u to v and we denote such a graph by $\overrightarrow{G}$.

In the above definition, note that v and u in the set E are not necessarily distinct. So loops are allowed which are vertices with an edge from the vertex to itself.

Definition 4.

Let $\overrightarrow{G}=(V,E)$ be a directed graph and let $v,w\in V$. We define the relation $v{E}_{{(l)}_i}^{n_i}w$ for $i\in \mathbf{N}$ to be the edges directed from the vertex v to vertex w where l is the number of edges starting from v and $n_i$ is the length of each $l_i$ path from v to w.

Definition 5.

Let $\overrightarrow{G}=(V,E)$ be a directed graph. Then $\overrightarrow{G}$ is called a graph of l-arity and denoted by ${\overrightarrow{G}}_l$ if the following holds:

$$\overrightarrow{G}\vDash \forall v[{\exists }^{=l}w \mathit{such} \mathit{that} v{E}_{l_{(i)}}^{n_i}w.$$

If $l=1$ then ${\overrightarrow{G}}_1$ is called a graph of 1-arity and is satisfies the formula

$$\forall v{\exists }^{=1}w \mathit{such} \mathit{that} v{E}_1^{n}w.$$

Definition 6.

Let ${\overrightarrow{G}}_1=(V,E)$ be a directed graph with 1-arity. Then ${\overrightarrow{G}}_1=(V,E)$ is called a single graph if it satisfy the following

$$\forall v[{\exists }^{\le 1}u \mathit{such} \mathit{that} uEv],$$

we denote such a graph by $Sing({\overrightarrow{G}}_1)$.

If $v\in V$ such that v satisties the formula

$${\exists }^{=0}u \mathit{such} \mathit{that} uEv],$$

then v is the zero vertex and denoted by $v_0$.

2. Classification of Directed Single Graphs of 1-Arity

Definition 7.

Let $C\subseteq V$ and $v,w\in V$. If there is $n\in \mathbb{N}$ such that $v{E}^{n}w$ then $v,w$ are connected. Let $C\subseteq V$. If any two vertices of C are connected then we say that C is connected. A connected component of $Sing({\overrightarrow{G}}_1)$ is maximal connected graph.

Lemma 1.

Every connected component of $Sing({\overrightarrow{G}}_1)$ is either

1. 

a copy of $C_0=\{v_0{E}^{n}w:\mathit{for} \mathit{each} n\in \mathbb{N}\}$,

2. 

a copy of $C_{\omega }=\{v{E}^{n}w:\mathit{for} \mathit{each} n\in \mathbb{Z}\}$,

3. 

a copy of $C_r=\{v_i{E}^r{v}_i:\mathit{for} \mathit{some} r\in {\mathbb{N}}^{+}\}$.

Proof. 

Let C be a finite connected component of $Sing({\overrightarrow{G}}_1)$. Let $v\in C$. Then

$$v{E}^{0}v,v{E}^1{v}_1,v{E}^2{v}_2,v{E}^3{v}_3,\dots \in C.$$

So $v{E}^n{v}_i\in C$ for all $n\in \mathbb{N}$. Let r be the smallest number in $\mathbb{N}$ such that $v{E}^r{v}_i=v{E}^q{v}_i$ for some $q\in \{0,\dots ,r-1\}$. If $q\ne 0$, $v{E}^{r-1}{v}_i=v{E}^{q-1}{v}_i$ as $Sing({\overrightarrow{G}}_1)$ is a directed single graph with 1-arity. So, $q-1\in \{0,\dots ,r-2\}$, contradicting that r is the smallest such number. So $q=0$ which means that C contains a cycle of period r. Since $Sing({\overrightarrow{G}}_1)$ is a directed single graph with 1-arity, $C=C_r$.

Now let C be infinite connected component. Let $v\in C$. Then we have

$$v{E}^{0}v,v{E}^1{v}_1,v{E}^2{v}_2,v{E}^3{v}_3,\dots ,v{E}^n{v}_i,\dots \mathit{for}n\in \mathbb{N}.$$

If $v{E}^r{v}_i=v{E}^q{v}_i$ for some $r,q\in \mathbb{N}$, and $r\ne q$, then C is finite by the previous argument. So all $v{E}^n{v}_i$ are distinct for $n\in \mathbb{N}$ and we will have two possibilities:

First, there is $v_0\in C$ such that $v_0$ satisfies the formula

$${\exists }^{=0}w \mathit{such} \mathit{that} wEv,$$

then C is $\{v_0,v_0{E}^1{v}_1,v_0{E}^2{v}_2,\dots ,v_0{E}^n{v}_n,\dots \}$, i.e., $C=\{v_0{E}^{n}w: \mathit{for} \mathit{each} n\in \mathbb{Z},v\}=C_0$.

Second, choosing any $v\in C$, such that v satisfies the formula

$${\exists }^{=1}w \mathit{such} \mathit{that} wEv$$

and w is unique. So we have $v{E}^{n}w\in C$ for each $n\in \mathbb{Z}$ and for any $n,m\in \mathbb{Z}$, $v{E}^{n}w$ is distinct from $v{E}^{m}w$ unless $n=m$. So, $C=C_{\omega }$. □

Example 1.

1. 

Consider the structure $(\mathbb{Q},<)$ where the vertices are $q\in \mathbb{Q}$ and the edges defined regarding the relation < as follows:

$$\forall q_i[q_{i}E{q}_j\iff q_i<q_j\wedge (\neg \exists q_r \mathit{such} \mathit{that} q_i<q_r<q_j)].$$

This formula guarantee there is exactly one edge going from $q_i$ to $q_j$. This formula is also true in $(\mathbb{Z},<)$ but not true in $(\mathbb{R},<)$ as there is always an element $q_r\in \mathbb{R}$ such that $q_i<q_r<q_j$. We can consider $(\mathbb{Q},<)$ and $(\mathbb{Z},<)$ regarding the above formula as copies of the connected component $C_{\omega }$ in $Sing({\overrightarrow{G}}_1)$. From this example, we learned a mathematical property about $(\mathbb{Q},<)$, $(\mathbb{Z},<)$, and $(\mathbb{R},<)$ which is $(\mathbb{Q},<)$, $(\mathbb{Z},<)$ are countable structure and $(\mathbb{R},<)$ is uncountable.

2. 

Consider the structure $(\mathbb{N},<)$, then we need to the formula above will be:

$$\forall q_i[q_{i}E{q}_j\iff q_i<q_j\wedge (\neg \exists q_r \mathit{such} \mathit{that} q_i<q_r<q_j)\wedge ({\exists }^{=0}{q}_k \mathit{such} \mathit{that} q_k<q_0)].$$

We can consider $(\mathbb{N},<)$ regarding the above formula as copy of the connected component $C_0$ in $Sing({\overrightarrow{G}}_1)$.

3. 

Consider the permutation On $A_4=\{1,2,3,4\}$ defined by

$$\sigma =\left(\begin{array}{cccc}1& 2& 3& 4\\ 3& 4& 2& 1\end{array}\right)$$

It is defined by the formula

$$iEj\iff \sigma (i)=j.$$

This permutation can be considered as a copy of the connected component $C_4$ in $Sing({\overrightarrow{G}}_1)$. Now take another permutation on $A_4$ defined by

$$\tau =\left(\begin{array}{cccc}1& 2& 3& 4\\ 2& 1& 4& 3\end{array}\right)$$

It is also defined by the formula

$$iEj\iff \tau (i)=j.$$

By looking to this formula defined on τ we can see we have two connected components of $C_2$ in $Sing({\overrightarrow{G}}_1)$. The permutations σ and τ are in the permutation group $S_4$ of $A_4$ but from the formulas above we know what is the algebraic structures of these two permutations.

Such methodology can be applied on many algebraic structures, for example $({\mathbb{Z}}_4,+)$ and hence we can study the cyclic subgroups in ${\mathbb{Z}}_4$.

4. 

Ref. [7] Consider the structure $(\mathbb{N},Succ)$ where $Succ$ is the successor function defined on the natural numbers as $S(n)=n+1$. It can be defined by the formula

$$\forall n_i[n_{i}E{n}_j\iff S(n_i)=n_j\wedge ({\exists }^{=0}{n}_k \mathit{such} \mathit{that} S(n_k)=0)].$$

We can consider $(\mathbb{N},Succ)$ regarding the above formula as copy of the connected component $C_0$ in $Sing({\overrightarrow{G}}_1)$.

Definition 8.

Let $Sing({\overrightarrow{G}}_1)=\langle V,E\rangle$. We define the characteristic sequence for $Sing({\overrightarrow{G}}_1)$ to be

$$\rho (Sing({\overrightarrow{G}}_1))={({\rho }_r)}_{r\in \mathbb{N}\cup \left\{\omega \right\}}$$

a sequence of cardinals such that ${\rho }_0$ is the numbers of copies of $C_0$, ${\rho }_{\omega }$ is the numbers of copies of $C_{\omega }$, and ${\rho }_r$ is the number of copies of $C_r$ for $r\in {\mathbb{N}}^{+}$.

Proposition 1.

Let $Sing({\overrightarrow{G}}_1)=\langle V,E\rangle$ and $Sing({\overrightarrow{H}}_1)=\langle W,E\rangle$. If $\rho (Sing({\overrightarrow{G}}_1))=$$\rho (Sing({\overrightarrow{H}}_1))$ Then $Sing({\overrightarrow{G}}_1)\cong Sing({\overrightarrow{H}}_1)$.

2.1. Axiomatization of the Theory of Directed Single Graphs of 1-Arity

Definition 9.

For a sequence $\rho ={({\rho }_r)}_{r\in \mathbb{N}}$ where each ${\rho }_r\in \mathbb{N}\cup \left\{\omega \right\}$, we axiomatize the theory $T{h}_{\rho }$ by the following:

$$\forall v[{\exists }^{=1}w \mathit{such} \mathit{that} vEw].$$

$$\forall v[{\exists }^{\le 1}w \mathit{such} \mathit{that} wEv].$$

If ${\rho }_0$ is finite, then

$${\exists }^{={\rho }_0}v[{\exists }^{=0}w[wEv]].$$

If ${\rho }_0$ is infinite, then

$${\exists }^{\ge n}v[{\exists }^{=0}w[wEv]] \mathit{for} \mathit{each} n\in {\mathbb{N}}^{+}.$$

For each $r\in {\mathbb{N}}^{+}$, if ${\rho }_r$ is finite, then

$${\exists }^{=n}v[v{E}^{r}v\wedge \underset{q|r,q\ne r}{\bigwedge }\neg (v{E}^{q}v)]\mathit{where}n={\rho }_r.r.$$

For each $r\in {\mathbb{N}}^{+}$, if ${\rho }_r$ is infinite, then

$${\exists }^{\ge n}v[v{E}^{r}v\wedge \underset{q|r,q\ne r}{\bigwedge }\neg (v{E}^{q}v)] \mathit{for} \mathit{each} n\in {\mathbb{N}}^{+}.$$

If $\forall v[{\exists }^{=1}w \mathit{such} \mathit{that} wEv]$ and there is $N\in \mathbb{N}$ such that for all $r>N$, $\neg [v{E}^{r}v\wedge {\displaystyle \underset{q|r,q\ne r}{\bigwedge }}\neg (v{E}^{q}v)]$ and ${\displaystyle \sum _{r=1}^N}{\rho }_r$ is infinite, then

$$\forall v\underset{i=1}{\overset{N}{\bigvee }}[v{E}^{i}v\wedge \underset{q|i,q\ne i}{\bigwedge }\neg (v{E}^{q}v)].$$

For each $n\in {\mathbb{N}}^{+}$,

$${\exists }^{\ge n}v[v=v].$$

Definition 10.

We define the dominant component in $Sing({\overrightarrow{G}}_1)$ to be the component $C_i$ for some $i\in \mathbb{N}$ such that the following holds:

$$|{\rho }_i{C}_i|>|{\rho }_j{C}_j| \mathit{for} \mathit{all} j\in \mathbb{N} \mathit{and} j\ne i.$$

If the dominant component in $Sing({\overrightarrow{G}}_1)$ is $C_i$ for some $i\in {\mathbb{N}}^{+}$, then $Sing({\overrightarrow{G}}_1)$ is with cyclic dominant component of length i. Otherwise, $Sing({\overrightarrow{G}}_1)$ is with the infinite dominant component $C_0$.

Definition 11.

We define $char(\rho )(Sing({\overrightarrow{G}}_1))$ to be the characteristic sequence for strongly minimal directed single graphs of 1-arity in the language $L=\langle E\rangle$.

Lemma 2.

If $Sing({\overrightarrow{G}}_1)=\langle V,E\rangle$ is strongly minimal, then the characteristic sequence $char(\rho (Sing({\overrightarrow{G}}_1))$ satisfies one of the following cases:

Case 1:

There is only finitely many infinite dominant components, that is, $|{\rho }_0.v_0|$ is finite.

Case 2:

If $Sing({\overrightarrow{G}}_1)$ is with cyclic dominant component of length i for some $i\in {\mathbb{N}}^{+}$, then $|{\rho }_i{C}_i|$ is infinite, $|{\rho }_0{C}_0|=0$, $|{\rho }_{\omega }{C}_{\omega }|=0$, and ${\displaystyle \sum _{q\ne i}}{\rho }_q.C_q$ is finite.

Proof. 

Case 1:

Let $\theta (v)$ be the formula

$${\exists }^{=1}w[vEw]$$

As $Sing({\overrightarrow{G}}_1)$ is 1-arity, $|\theta (v),Sing({\overrightarrow{G}}_1)|$ is infinite which means that $\theta (v)$ defines an infinite subset of $Sing({\overrightarrow{G}}_1)$. Now, $\neg \theta (v)$ will be the formula

$$\neg {\exists }^{=1}w[vEw]$$

which is equivalent to the formula

$${\exists }^{=0}w[vEw].$$

So $\neg \theta (v)$ defines the zero vertices $v_0$ in $Sing({\overrightarrow{G}}_1)$. As $Sing({\overrightarrow{G}}_1)$ is strongly minimal, $\neg \phi (v)$ defines a finite set. Hence, there is only finitly many $v_0$ in $Sing({\overrightarrow{G}}_1)$. So there is only finitely many infinite dominant component.

Case 2:

Suppose $Sing({\overrightarrow{G}}_1)$ is with cyclic dominant component of length i for some $i\in {\mathbb{N}}^{+}$. Then

$$|{\rho }_i{C}_i|>|{\rho }_j{C}_j| \mathit{for} \mathit{all} j\in \mathbb{N} \mathit{and} j\ne i.$$

Let $\phi (\overline{a},v)$ be the formula $v{E}^{i}v\wedge {\displaystyle \underset{q|r,q\ne i}{\bigwedge }}\neg (v{E}^{q}v)$. Now, $|{\rho }_0{C}_0|={\rho }_0.{\aleph }_0$ and $|{\rho }_{\omega }{C}_{\omega }|={\rho }_{\omega }.{\aleph }_0$. But $C_i$ is a cyclic dominant component. So

$$|{\rho }_i{C}_i|>{\rho }_0.{\aleph }_0$$

and

$$|{\rho }_i{C}_i|>{\rho }_{\omega }.{\aleph }_0.$$

Thus, $|\phi (\overline{a},Sing({\overrightarrow{G}}_1))|$ is infinite which means there is infinitely many cyclic component of length i and $\phi (\overline{a},Sing({\overrightarrow{G}}_1))$ defines this infinite subset of $Sing({\overrightarrow{G}}_1)$. Since $Sing({\overrightarrow{G}}_1)$ is strongly minimal, $\neg (\phi (\overline{a},Sing({\overrightarrow{G}}_1)))$ defines a finite subset of $Sing({\overrightarrow{G}}_1)$. Hence, $|{\rho }_0{C}_0|=0$, $|{\rho }_{\omega }{C}_{\omega }|=0$, and ${\displaystyle \sum _{q\ne i}}{\rho }_q.q$ is finite.

□

2.2. Models of the Theory of Strongly Minimal Directed Single Graphs of 1-Arity

Proposition 2.

The theory of strongly minimal directed single graphs of 1-arity $T{h}_{char(\rho )}$ satisfies the following:

1. 

If the dominant component is cyclic of length i for some $i\in {\mathbb{N}}^{+}$ then $T{h}_{char(\rho )}$ is totally categorical

2. 

If the dominant component is infinite then $T{h}_{char(\rho )}$ is uncountably categorical

Proof. 

• Suppose that the dominant component is cyclic of length i for some $i\in {\mathbb{N}}^{+}$. Then by Theorem 2, $|{\rho }_0{(C_0)}_{char(\rho )}|=0$, and $|{\rho }_{\omega }{(C_{\omega })}_{char(\rho )}|=0$, and $|{\rho }_i{(C_i)}_{char(\rho )}|$ is infinite, and ${\displaystyle \sum _{q\ne i}}{\rho }_q.{(C_q)}_{char(\rho )}$ is finite. This means that there is $N\in \mathbb{N}$ such that if $r>N$ then $|{\rho }_r.{(C_r)}_{char(\rho )}|=0$. Hence, $T{h}_{char(\rho )}$ satisfy the formula

  $$\forall v\underset{j=1}{\overset{N}{\bigvee }}[v{E}^{j}v\wedge \underset{q|j,q\ne j}{\bigwedge }\neg (v{E}^{q}v)].$$

  Suppose ${Sing({\overrightarrow{G}}_1)}_{\kappa }\vDash T{h}_{char(\rho )}$. Then, $|{Sing({\overrightarrow{G}}_1)}_{\kappa }|$ will be

  $$|{\rho }_0.{(C_0)}_{char(\rho )}|+\sum _{q\ne r}q·|{\rho }_q.{(C_q)}_{char(\rho )}|+\kappa ·r+|{\rho }_{\omega }{(C_{\omega })}_{char(\rho )}|=\kappa$$

  as $\kappa =|{\rho }_r.(Sing({\overrightarrow{G}}_1))|$. Therefor, $T{h}_{char(\rho )}$ is totally categorical.
• Suppose the dominant component is infinite. Then $|{\rho }_0.{(v_0)}_{char(\rho )}|>0$. So $|{\rho }_q.{(C_q)}_{char(\rho )}|$ is finite for all $q\in {\mathbb{N}}^{+}$. Suppose ${Sing({\overrightarrow{G}}_1)}_{\kappa }\vDash T{h}_{char(\rho )}$. Then, $|{Sing({\overrightarrow{G}}_1)}_{\kappa }|$ will be

  $${\aleph }_0·|{\rho }_0.{(v_0)}_{char(\rho )}|+\sum _{q\in {\mathbb{N}}^{+}}q·|{\rho }_q.{(C_q)}_{char(\rho )}|+\kappa ·{\aleph }_0={\aleph }_0+\kappa$$

  where $\kappa =|{\rho }_{\omega }.{(C_{\omega })}_{(Sing({\overrightarrow{G}}_1))}|$.
  Therefor, $T{h}_{char(\rho )}$ is uncountably categorical.

□

Proposition 3.

The theory of strongly minimal directed single graphs of 1-arity $T{h}_{char(\rho )}$ is complete.

Proof. 

• If $|{\rho }_0.{(v_0)}_{char(\rho )}|=0$ and ${\displaystyle \sum _{q\in {\mathbb{N}}^{+}}}q.|{\rho }_q.{(C_q)}_{char(\rho )}|$ is finite for all $q\in {\mathbb{N}}^{+}$. So there is $N\in \mathbb{N}$ such that if $q>N$ then $|{\rho }_q.{(C_q)}_{char(\rho )}|=0$. So, $T{h}_{char(\rho )}$ satisfy the formula

  $$\forall v\underset{q=1}{\overset{N}{\bigvee }}[v{E}^{q}v\wedge \underset{s|q,s\ne q}{\bigwedge }\neg (v{E}^{s}v)].$$
  This means that if $Sing({\overrightarrow{G}}_1)\vDash T{h}_{char(\rho )}$ then $|{\rho }_q.{(C_q)}_{Sing({\overrightarrow{G}}_1)}|$ is finite. So the complement of this set is not empty as $Sing({\overrightarrow{G}}_1)$ is infinite. So there is at least on copy of ${(C_{\omega })}_{Sing({\overrightarrow{G}}_1)}$. Suppose ${Sing({\overrightarrow{G}}_1)}_{\kappa }\vDash T{h}_{char(\rho )}$. Then, $|{Sing({\overrightarrow{G}}_1)}_{\kappa }|$ will be

  $${\aleph }_0·|{\rho }_0.{(v_0)}_{char(\rho )}|+\sum _{q\in {\mathbb{N}}^{+}}q·|{\rho }_q.{(C_q)}_{char(\rho )}|+\kappa ·{\aleph }_0={\aleph }_0+\kappa$$

  where $\kappa =|{\rho }_{\omega }.{(C_{\omega })}_{(Sing({\overrightarrow{G}}_1))}|$.
• If $|{\rho }_0.{(v_0)}_{char(\rho )}|=0$ and $|{\rho }_q.{(C_q)}_{char(\rho )}|$ is finite for all $q\in {\mathbb{N}}^{+}$ but $\sum _{q\in {\mathbb{N}}^{+}}q.|{\rho }_q.{(C_q)}_{char(\rho )}|$ is infinite, then $|{Sing({\overrightarrow{G}}_1)}_{\kappa }|$ will be

  $${\aleph }_0·|{\rho }_0.{(v_0)}_{char(\rho )}|+\sum _{q\in {\mathbb{N}}^{+}}q·|{\rho }_q.{(C_q)}_{char(\rho )}|+\kappa ·{\aleph }_0={\aleph }_0+\kappa$$

  where $\kappa =|{\rho }_{\omega }.{(C_{\omega })}_{(Sing({\overrightarrow{G}}_1))}|$.
  So from the above cases and from Proposition 2, $T{h}_{char(\rho )}$ is uncountably categorical. Also from the axiomatization of $char(\rho )$, any model of $T{h}_{cha{r}_{\rho }}$ satisfy the axiom

  $${\exists }^{\ge n}v[v=v] \mathit{for} \mathit{each} n\in {\mathbb{N}}^{+}.$$

  Hence, by by the Łos-Vaught test, $T{h}_{char(\rho )}$ is complete.

□

3. Conclusions and Future Work

This research started with classifying $Sing({\overrightarrow{G}}_1)$ and then introduced all the axioms of the theory of $Sing({\overrightarrow{G}}_1)$. Then we proved the conditions of $char(\rho (Sing({\overrightarrow{G}}_1))$ for the strongly minimal $Sing({\overrightarrow{G}}_1)$ and examined the theory $T{h}_{char(\rho )}$ and proved is complete. The importance of such theory of such graphs is that we can apply many mathematical structures on it where as algebraically closed fields narrowed only on field theory.For example, we can apply groups, permutation groups, and other mathematical sets with some relation such as ordering relation and the successor function.

The goal of this work is we will show in the future that these structures can be Zariski Geometry. Zariski axioms relate algebra to geometry by topologize strongly minimal structures. When this is done the work will even be extended to directed single graphs with l-arity not only 1-arity and give the complete theory of such graphs so that we can extend the range of the mathematical structure that can be applied on these graphs. Again we will show that these structures are Zariski Geometry. $\mathbf{Hrushovski}$ and $\mathbf{Zilber}$ classified strongly minimal structures which are Zariski Geometry are only algebraically closed fields, vector spaces, affine spaces, and pure sets [8] and algebraically closed field are on top of these classification. Thus this research is aiming to be extended in the future to find a wide range of mathematical structures which is not listed in $\mathbf{Hrushovski}$ and $\mathbf{Zilber}$ classification.