# Primeness of Relative Annihilators in BCK-Algebra

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

- (I)
- $(\forall x,y,z\in X)$$\left(\right((x\ast y)\ast (x\ast z))\ast (z\ast y)=0),$
- (II)
- $(\forall x,y\in X)$$\left(\right(x\ast (x\ast y))\ast y=0),$
- (III)
- $(\forall x\in X)$$(x\ast x=0),$
- (IV)
- $(\forall x,y\in X)$$(x\ast y=0,\phantom{\rule{3.33333pt}{0ex}}y\ast x=0\phantom{\rule{3.33333pt}{0ex}}\Rightarrow \phantom{\rule{3.33333pt}{0ex}}x=y).$

**Proposition**

**1.**

- (1)
- $(\forall x\in X)$$(x\ast 0=x),$
- (2)
- $(\forall x,y,z\in X)$$(x\le y\phantom{\rule{0.166667em}{0ex}}\Rightarrow \phantom{\rule{0.166667em}{0ex}}x\ast z\le y\ast z,\phantom{\rule{0.166667em}{0ex}}z\ast y\le z\ast x),$
- (3)
- $(\forall x,y,z\in X)$$\left(\right(x\ast y)\ast z=(x\ast z)\ast y),$
- (4)
- $(\forall x,y,z\in X)$$\left(\right(x\ast z)\ast (y\ast z)\le x\ast y)$

**Definition**

**1.**

**Definition**

**2**

**.**Let X be a a $BCI/BCK$-algebra. An arbitrary subset A of X is called an ideal of X if it satisfies

**Remark**

**1**

**.**For every ideal A of a $BCK$-algebra X and for all $x,y\in X$, the following implication is satisfied:

**Definition**

**3**

**.**Let P be a proper ideal of a lower $BCK$-semilattice X. Then, P is a prime ideal if, for $a,b\in X$ such that $a\wedge b\in P$, we conclude that $a\in P$ or $b\in P$, where $a\wedge b$ is the greatest lower bound of a and b.

**Lemma**

**1**

**.**If $\phi :X\to Y$ is an epimorphism of lower $BCK$-semilattices, then

**Lemma**

**2**

**.**1. Let $\phi :X\to Y$ be an epimorphism of $BCK$-algebras. If A is an ideal of X, then $\phi \left(A\right)$ is an ideal of Y.

**Lemma**

**3**

**.**Let $\phi :X\to Y$ be a homomorphism of BCK-algebras X and Y and let A be an ideal of X such that $Ker\left(\phi \right)\subseteq A$. Then, ${\phi}^{-1}\left({A}^{\prime}\right)=A$ where ${A}^{\prime}=\phi \left(A\right)$.

## 3. Primeness of Relative Annihilators

**Definition**

**4**

**.**Let A and B be two arbitrary subsets of X. A set $\left(A{:}_{\wedge}B\right)$ is defined as follows:

**Remark**

**2.**

**Lemma**

**4.**

**Lemma**

**5.**

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

**Lemma**

**6**

**.**If A and B are ideals of X, then the relative annihilator $\left(A{:}_{\wedge}B\right)$ of B with respect to A is an ideal of X.

**Theorem**

**2.**

**Proof.**

**Example**

**1.**

∗ | 0 | 1 | 2 | 3 | 4 |

0 | 0 | 0 | 0 | 0 | 0 |

1 | 1 | 0 | 0 | 1 | 1 |

2 | 2 | 2 | 0 | 2 | 2 |

3 | 3 | 3 | 3 | 0 | 3 |

4 | 4 | 4 | 4 | 4 | 0 |

**Lemma**

**7.**

**Proof.**

**Theorem**

**3.**

- (i)
- ${A}_{1}\subseteq P$ or ${A}_{2}\subseteq P$.
- (ii)
- ${A}_{1}\cap {A}_{2}\subseteq P$.
- (iii)
- ${A}_{1}\wedge {A}_{2}\subseteq P$.

**Proof.**

**Theorem**

**4.**

- (i)
- ${A}_{j}\subseteq P$ for some $j\in \{1,2,\cdots ,n\}$.
- (ii)
- ${\bigcap}_{i=1}^{n}{A}_{i}\subseteq P$.
- (iii)
- ${\bigwedge}_{i=1}^{n}{A}_{i}\subseteq P$.

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Definition**

**5.**

- (1)
- $A={\displaystyle \bigcap _{i\in \{1,2,\cdots ,n\}}}{Q}_{i}$,
- (2)
- $\bigcap _{\genfrac{}{}{0pt}{}{i\in \{1,2,\cdots ,n\}}{i\ne j}}}{Q}_{i}\u2288{Q}_{j$.

**Lemma**

**8**

**.**Let A, B, and C be non-empty subsets of X. Then, we have

**Theorem**

**7.**

- (i)
- $P={P}_{1}$ or $P={P}_{2}$.
- (ii)
- There exists $a\in X$ such that $\left(A{:}_{\wedge}a\right)=P$.

**Proof.**

**Theorem**

**8.**

- (i)
- $P={P}_{i}$ for some $i\in \{1,2,\cdots ,n\}$.
- (ii)
- There exists $a\in X$ such that $\left(A{:}_{\wedge}a\right)=P$.

**Theorem**

**9.**

- (i)
- If P is a prime ideal of X such that $Ker\phi \subseteq P$, then $\phi \left(P\right)$ is a prime ideal of Y.
- (ii)
- For prime ideals ${P}_{1},{P}_{2},\cdots ,{P}_{n}$ of X, the following equation is satisfied:$$\phi ({P}_{1}\cap {P}_{2}\cap \cdots \cap {P}_{n})=\phi \left({P}_{1}\right)\cap \phi \left({P}_{2}\right)\cap \cdots \cap \phi \left({P}_{n}\right).$$

**Proof.**

**Lemma**

**9.**

**Proof.**

**Theorem**

**10.**

**Proof.**

**Corollary**

**2.**

**Theorem**

**11.**

**Proof.**

**Lemma**

**10**

**.**If X is Noetherian $BCK$-algebra, then each ideal of X has a unique minimal prime decomposition.

**Lemma**

**11**

**.**Every proper ideal of X is equal to the intersection of all minimal prime ideals associated with it.

**Example**

**2.**

∗ | 0 | 1 | 2 | 3 | 4 |

0 | 0 | 0 | 0 | 0 | 0 |

1 | 1 | 0 | 0 | 0 | 0 |

2 | 2 | 1 | 0 | 1 | 1 |

3 | 3 | 3 | 3 | 0 | 3 |

4 | 4 | 4 | 4 | 4 | 0 |

**Example**

**3.**

∗ | 0 | 1 | 2 | 3 | 4 |

0 | 0 | 0 | 0 | 0 | 0 |

1 | 1 | 0 | 1 | 0 | 1 |

2 | 2 | 2 | 0 | 2 | 0 |

3 | 3 | 1 | 3 | 0 | 3 |

4 | 4 | 4 | 4 | 4 | 0 |

**Theorem**

**12.**

- (i)
- P is a prime ideal of X.
- (ii)
- $X\backslash P$ is a closed subset under the ∧ operation in X, that is, $x\wedge y\in X\backslash P$ for all $x,y\in X\backslash P$.

**Proof.**

**Definition**

**6.**

**Definition**

**7**

- (i)
- “$cl$” is a semi-prime closure operation if we have$$A\wedge {B}^{cl}\subseteq {(A\wedge B)}^{cl}and{A}^{cl}\wedge B\subseteq {(A\wedge B)}^{cl}$$
- (ii)
- “$cl$” is a good semi-prime closure operation, if we have$$A\wedge {B}^{cl}={A}^{cl}\wedge B={(A\wedge B)}^{cl}$$

**Theorem**

**13**

**.**Suppose that “$cl$” is a semi-prime closure operation on X and S is a closed subset of X under the ∧ operation. If X is Noetherian and A is a $cl$-closed ideal of X, then the set

**Lemma**

**12.**

**Proof.**

**Theorem**

**14.**

**Proof.**

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**MDPI and ACS Style**

Bordbar, H.; Muhiuddin, G.; Alanazi, A.M.
Primeness of Relative Annihilators in *BCK*-Algebra. *Symmetry* **2020**, *12*, 286.
https://doi.org/10.3390/sym12020286

**AMA Style**

Bordbar H, Muhiuddin G, Alanazi AM.
Primeness of Relative Annihilators in *BCK*-Algebra. *Symmetry*. 2020; 12(2):286.
https://doi.org/10.3390/sym12020286

**Chicago/Turabian Style**

Bordbar, Hashem, G. Muhiuddin, and Abdulaziz M. Alanazi.
2020. "Primeness of Relative Annihilators in *BCK*-Algebra" *Symmetry* 12, no. 2: 286.
https://doi.org/10.3390/sym12020286