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

## Abstract

**:**

## 1. Introduction and Preliminaries

**Definition**

**1**

**.**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**

**.**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**

**.**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**

**.**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.**

**Definition**

**4.**

**Definition**

**5.**

**Definition**

**6.**

## 2. Classification of Directed Single Graphs of 1-Arity

**Definition**

**7.**

**Lemma**

**1.**

- 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.**

**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.**

**Proposition**

**1.**

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

**Definition**

**9.**

**Definition**

**10.**

**Definition**

**11.**

**Lemma**

**2.**

- 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 $\sum _{q\ne i}}{\rho}_{q}.{C}_{q$ 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$$\neg {\exists}^{=1}w[vEw]$$$${\exists}^{=0}w[vEw].$$
- Case 2:
- Suppose $Sing({\overrightarrow{G}}_{1})$ is with cyclic dominant component of length i for some $i\in {\mathbb{N}}^{+}$. Then$$|{\rho}_{i}{C}_{i}|>|{\rho}_{j}{C}_{j}|\mathit{for}\mathit{all}j\in \mathbb{N}\mathit{and}j\ne i.$$$$|{\rho}_{i}{C}_{i}|>{\rho}_{0}.{\aleph}_{0}$$$$|{\rho}_{i}{C}_{i}|>{\rho}_{\omega}.{\aleph}_{0}.$$

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

**Proposition**

**2.**

- 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 $\sum _{q\ne i}}{\rho}_{q}.{({C}_{q})}_{char(\rho )$ 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$$\forall v\underset{j=1}{\overset{N}{\bigvee}}[v{E}^{j}v\wedge \underset{q|j,q\ne j}{\bigwedge}\neg (v{E}^{q}v)].$$$$|{\rho}_{0}.{({C}_{0})}_{char(\rho )}|+\sum _{q\ne r}q\xb7|{\rho}_{q}.{({C}_{q})}_{char(\rho )}|+\kappa \xb7r+|{\rho}_{\omega}{({C}_{\omega})}_{char(\rho )}|=\kappa $$
- 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$${\aleph}_{0}\xb7|{\rho}_{0}.{({v}_{0})}_{char(\rho )}|+\sum _{q\in {\mathbb{N}}^{+}}q\xb7|{\rho}_{q}.{({C}_{q})}_{char(\rho )}|+\kappa \xb7{\aleph}_{0}={\aleph}_{0}+\kappa $$Therefor, $T{h}_{char(\rho )}$ is uncountably categorical.

**Proposition**

**3.**

**Proof.**

- If $|{\rho}_{0}.{({v}_{0})}_{char(\rho )}|=0$ and $\sum _{q\in {\mathbb{N}}^{+}}}q.|{\rho}_{q}.{({C}_{q})}_{char(\rho )}|$ 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$${\aleph}_{0}\xb7|{\rho}_{0}.{({v}_{0})}_{char(\rho )}|+\sum _{q\in {\mathbb{N}}^{+}}q\xb7|{\rho}_{q}.{({C}_{q})}_{char(\rho )}|+\kappa \xb7{\aleph}_{0}={\aleph}_{0}+\kappa $$
- 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$${\aleph}_{0}\xb7|{\rho}_{0}.{({v}_{0})}_{char(\rho )}|+\sum _{q\in {\mathbb{N}}^{+}}q\xb7|{\rho}_{q}.{({C}_{q})}_{char(\rho )}|+\kappa \xb7{\aleph}_{0}={\aleph}_{0}+\kappa $$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$${\exists}^{\ge n}v[v=v]\mathit{for}\mathit{each}n\in {\mathbb{N}}^{+}.$$

## 3. Conclusions and Future Work

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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Albalahi, A.M.
Characteristic Sequence of Strongly Minimal Directed Single Graphs of 1-Arity. *Computation* **2022**, *10*, 220.
https://doi.org/10.3390/computation10120220

**AMA Style**

Albalahi AM.
Characteristic Sequence of Strongly Minimal Directed Single Graphs of 1-Arity. *Computation*. 2022; 10(12):220.
https://doi.org/10.3390/computation10120220

**Chicago/Turabian Style**

Albalahi, Abeer M.
2022. "Characteristic Sequence of Strongly Minimal Directed Single Graphs of 1-Arity" *Computation* 10, no. 12: 220.
https://doi.org/10.3390/computation10120220