# On a Graph Associated to UP-Algebras

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

**Example**

**5.**

**Example**

**6.**

**Proposition**

**1.**

- (1)
- $x\ast x=0$,
- (2)
- $x\ast y=0$ and $y\ast z=0\Rightarrow x\ast z=0,$
- (3)
- $x\ast y=0\Rightarrow (z\ast x)\ast (z\ast y)=0,$
- (4)
- $x\ast y=0\Rightarrow (y\ast z)\ast (x\ast z)=0,$
- (5)
- $x\ast (y\ast x)=0,$
- (6)
- $(y\ast x)\ast x=0\u27fax=y\ast x$ and
- (7)
- $x\ast (y\ast y)=0$

**Proposition**

**2.**

- (1)
- $x\le x$,
- (2)
- $x\le y$ and $y\le x\Rightarrow x=y,$
- (3)
- $x\le y$ and $y\le z\Rightarrow x\le z,$
- (4)
- $x\le y\Rightarrow z\ast x\le z\ast y,$
- (5)
- $x\le y\Rightarrow y\ast z\le x\ast z,$
- (6)
- $x\le y\ast x,$ and
- (7)
- $x\le y\ast y.$

**Proposition**

**3.**

**Proposition**

**4.**

- (1)
- $x\ast (y\ast z)=y\ast (x\ast z).$
- (2)
- $\left(\right(y\ast x)\ast x)\le y.$

**Definition**

**2.**

**Definition**

**3.**

- (i)
- The constant zero of A is in B and
- (ii)
- for ant$x,y,z\in A,$ $x\ast (y\ast z)\in B$ and $y\in B\Rightarrow x\ast z\in B.$

**Example**

**7.**

**Definition**

**4.**

**Theorem**

**1.**

- (1)
- A is commutative,
- (2)
- $(b\ast a)\ast a\le (a\ast b)\ast b,$
- (3)
- $\left(\right(a\ast b)\ast b)\ast \left(\right(b\ast a)\ast a)=0.$

**Proof.**

**Lemma**

**1.**

**Proof.**

**Definition**

**5.**

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

- (1)
- If $B\subseteq C,$ then $ann\left(C\right)\subseteq ann\left(B\right);$
- (2)
- $ann(B\cup C)=ann\left(B\right)\cap ann\left(C\right);$
- (3)
- $ann\left(B\right)\cup ann\left(C\right)\subseteq ann(B\cap C).$

**Proof.**

**Lemma**

**4.**

**Proof.**

**Definition**

**6.**

**Lemma**

**5.**

## 3. Graphs of Commutative UP-Algebras

**Definition**

**7.**

**Theorem**

**2.**

**Proof.**

**Example**

**8.**

Algorithm 1: Algorithm for UP-algebras. |

## 4. Graph of Equivalence Classes of a Commutative UP-Algebra A

**Lemma**

**6.**

**Proof.**

**Definition**

**8.**

**Example**

**9.**

**Lemma**

**7.**

- (1)
- ${G}_{E}\left(A\right)$ is a subgraph of $G\left(A\right);$
- (2)
- If $N\left(\right[0\left]\right)=A-\left\{0\right\},\phantom{\rule{0.277778em}{0ex}}\forall a\in A,$ then $G\left(A\right)$ is a star graph.

**Proof.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Lemma**

**8.**

**Proof.**

**Theorem**

**6.**

**Proof.**

**Example**

**10.**

## 5. Graph Folding

**Definition**

**9.**

- (1)
- For each vertex $v\in V\left({G}_{1}\right),F\left(v\right)$ is a vertex in $V\left({G}_{2}\right);$
- (2)
- For each edge $e\in E\left({G}_{1}\right),$ dim$\left(F\right(e\left)\right)\le dim\left(e\right).$

**Example**

**11.**

**Theorem**

**7.**

**Proof.**

**Corollary**

**1.**

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**MDPI and ACS Style**

Ansari, M.A.; Haidar, A.; Koam, A.N.A.
On a Graph Associated to UP-Algebras. *Math. Comput. Appl.* **2018**, *23*, 61.
https://doi.org/10.3390/mca23040061

**AMA Style**

Ansari MA, Haidar A, Koam ANA.
On a Graph Associated to UP-Algebras. *Mathematical and Computational Applications*. 2018; 23(4):61.
https://doi.org/10.3390/mca23040061

**Chicago/Turabian Style**

Ansari, Moin A., Azeem Haidar, and Ali N.A. Koam.
2018. "On a Graph Associated to UP-Algebras" *Mathematical and Computational Applications* 23, no. 4: 61.
https://doi.org/10.3390/mca23040061