# Commutative Generalized Neutrosophic Ideals in BCK-Algebras

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

- (I)
- $((x\ast y)\ast (x\ast z))\ast (z\ast y)=0,$
- (II)
- $(x\ast (x\ast y))\ast y=0,$
- (III)
- $x\ast x=0,$
- (IV)
- $x\ast y=0,\phantom{\rule{0.166667em}{0ex}}y\ast x=0\Rightarrow x=y.$

- (V)
- $0\ast x=0,$

**Lemma**

**1**

**([7]).**Let I be an ideal of a $BCK$-algebra X. Then I is commutative ideal of X if and only if it satisfies the following condition for all $x,y$ in X:

## 3. Commutative Generalized Neutrosophic Ideals

**Definition**

**1.**

**Example**

**1.**

**Theorem**

**1.**

**Proof.**

**Example**

**2.**

**Theorem**

**2.**

**Proof.**

**Lemma**

**2**

**([15]).**Any generalized neutrosophic ideal $A=({A}_{T},$ ${A}_{IT},$ ${A}_{IF},$ ${A}_{F})$ of X satisfies:

**Theorem**

**3.**

**Proof.**

**Lemma**

**3.**

**([15])**If a GNS $A=({A}_{T},$ ${A}_{IT},$ ${A}_{IF},$ ${A}_{F})$ in X is a generalized neutrosophic ideal of X, then the sets ${U}_{A}(T,{\alpha}_{T})$, ${U}_{A}(IT,{\alpha}_{IT})$, ${L}_{A}(F,{\beta}_{F})$ and ${L}_{A}(IF,{\beta}_{IF})$ are ideals of X for all ${\alpha}_{T}$, ${\alpha}_{IT}$, ${\beta}_{F}$, ${\beta}_{IF}\in [0,1]$ whenever they are non-empty.

**Theorem**

**4.**

**Proof.**

**Lemma**

**4**

**([15]).**Assume that $A=({A}_{T},$ ${A}_{IT},$ ${A}_{IF},$ ${A}_{F})$ is a GNS in X and ${U}_{A}(T,{\alpha}_{T})$, ${U}_{A}(IT,{\alpha}_{IT})$, ${L}_{A}(F,{\beta}_{F})$ and ${L}_{A}(IF,{\beta}_{IF})$ are ideals of X, $\forall {\alpha}_{T}$, ${\alpha}_{IT}$, ${\beta}_{F}$, ${\beta}_{IF}\in [0,1]$. Then $A=({A}_{T},$ ${A}_{IT},$ ${A}_{IF},$ ${A}_{F})$ is a generalized neutrosophic ideal of X.

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

- (1)
- $X=\bigcup _{t\in \mathsf{\Lambda}}{C}_{t}$,
- (2)
- $(\forall s,t\in \mathsf{\Lambda})\left(s>t\u27fa{C}_{s}\subset {C}_{t}\right)$

**Proof.**

- (i)
- $t=sup\{q\in \mathsf{\Lambda}\mid q<t\}$,
- (ii)
- $t\ne sup\{q\in \mathsf{\Lambda}\mid q<t\}$.

- (iii)
- $s=inf\{r\in \mathsf{\Lambda}\mid s<r\}$,
- (iv)
- $s\ne inf\{r\in \mathsf{\Lambda}\mid s<r\}$.

**Lemma**

**5**

**([15]).**Let $f:X\to Y$ be a homomorphism of $BCK/BCI$-algebras. If a GNS $A=({A}_{T},$ ${A}_{IT},$ ${A}_{IF},$ ${A}_{F})$ in Y is a generalized neutrosophic ideal of Y, then the new GNS ${A}^{f}=({A}_{T}^{f},$ ${A}_{IT}^{f},$ ${A}_{IF}^{f},$ ${A}_{F}^{f})$ in X is a generalized neutrosophic ideal of X.

**Theorem**

**8.**

**Proof.**

**Lemma**

**6**

**([15]).**Let $f:X\to Y$ be an onto homomorphism of $BCK/BCI$-algebras and let $A=({A}_{T},$ ${A}_{IT},$ ${A}_{IF},$ ${A}_{F})$ be a GNS in Y. If the induced GNS ${A}^{f}=({A}_{T}^{f},$ ${A}_{IT}^{f},$ ${A}_{IF}^{f},$ ${A}_{F}^{f})$ in X is a generalized neutrosophic ideal of X, then $A=({A}_{T},$ ${A}_{IT},$ ${A}_{IF},$ ${A}_{F})$ is a generalized neutrosophic ideal of Y.

**Theorem**

**9.**

**Proof.**

**Theorem**

**10.**

**Proof.**

**Theorem**

**11.**

**Proof.**

**Theorem**

**12.**

**Proof.**

**Theorem**

**13.**

**Proof.**

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Zhang, X.H. Fuzzy anti-grouped filters and fuzzy normal filters in pseudo-BCI algebras. J. Intell. Fuzzy Syst.
**2017**, 33, 1767–1774. [Google Scholar] [CrossRef] - Jun, Y.B. Neutrosophic subalgebras of several types in BCK/BCI-algebras. Ann. Fuzzy Math. Inform.
**2017**, 14, 75–86. [Google Scholar] - Jun, Y.B.; Kim, S.J.; Smarandache, F. Interval neutrosophic sets with applications in BCK/BCI-algebra. Axioms
**2018**, 7, 23. [Google Scholar] [CrossRef] - Jun, Y.B.; Smarandache, F.; Bordbar, H. Neutrosophic $\mathcal{N}$-structures applied to BCK/BCI-algebras. Information
**2017**, 8, 128. [Google Scholar] [CrossRef] - Jun, Y.B.; Smarandache, F.; Song, S.Z.; Khan, M. Neutrosophic positive implicative $\mathcal{N}$-ideals in BCK/BCI- algebras. Axioms
**2018**, 7, 3. [Google Scholar] [CrossRef] - Khan, M.; Anis, S.; Smarandache, F.; Jun, Y.B. Neutrosophic $\mathcal{N}$-structures and their applications in semigroups. Ann. Fuzzy Math. Inform.
**2017**, 14, 583–598. [Google Scholar] - Meng, J.; Jun, Y.B. BCK-Algebras; Kyung Moon Sa Co.: Seoul, Korea, 1994. [Google Scholar]
- Öztürk, M.A.; Jun, Y.B. Neutrosophic ideals in BCK/BCI-algebras based on neutrosophic points. J. Int. Math. Virtual Inst.
**2018**, 8, 1–17. [Google Scholar] - Saeid, A.B.; Jun, Y.B. Neutrosophic subalgebras of BCK/BCI-algebras based on neutrosophic points. Ann. Fuzzy Math. Inform.
**2017**, 14, 87–97. [Google Scholar] - Song, S.Z.; Smarandache, F.; Jun, Y.B. Neutrosophic commutative $\mathcal{N}$-ideals in BCK-algebras. Information
**2017**, 8, 130. [Google Scholar] [CrossRef] - Zhang, X.H.; Bo, C.X.; Smarandache, F.; Park, C. New operations of totally dependent- neutrosophic sets and totally dependent-neutrosophic soft sets. Symmetry
**2018**, 10, 187. [Google Scholar] [CrossRef] - Zhang, X.H.; Smarandache, F.; Liang, X.L. Neutrosophic duplet semi-group and cancellable neutrosophic triplet groups. Symmetry
**2017**, 9, 275. [Google Scholar] [CrossRef] - Huang, Y.S. BCI-Algebra; Science Press: Beijing, China, 2006. [Google Scholar]
- Smarandache, F. A Unifying Field in Logics. Neutrosophy: Neutrosophic Probability, Set and Logic; American Research Press: Rehoboth, NM, USA, 1999. [Google Scholar]
- Song, S.Z.; Khan, M.; Smarandache, F.; Jun, Y.B. A novel extension of neutrosophic sets and its application in BCK/BCI-algebras. In New Trends in Neutrosophic Theory and Applications (Volume II); Pons Editions; EU: Brussels, Belgium, 2018; pp. 308–326. [Google Scholar]
- Zhang, X.H.; Park, C.; Wu, S.P. Soft set theoretical approach to pseudo-BCI algebras. J. Intell. Fuzzy Syst.
**2018**, 34, 559–568. [Google Scholar] [CrossRef] - Zhang, X.H.; Bo, C.X.; Smarandache, F.; Dai, J.H. New inclusion relation of neutrosophic sets with applications and related lattice structure. Int. J. Mach. Learn. Cybern.
**2018**. [Google Scholar] [CrossRef]

∗ | 0 | a | b | c |
---|---|---|---|---|

0 | 0 | 0 | 0 | 0 |

a | a | 0 | 0 | a |

b | b | a | 0 | b |

c | c | c | c | 0 |

X | ${\mathit{A}}_{\mathit{T}}\left(\mathit{x}\right)$ | ${\mathit{A}}_{\mathit{IT}}\left(\mathit{x}\right)$ | ${\mathit{A}}_{\mathit{IF}}\left(\mathit{x}\right)$ | ${\mathit{A}}_{\mathit{F}}\left(\mathit{x}\right)$ |
---|---|---|---|---|

0 | $0.7$ | $0.6$ | $0.1$ | $0.3$ |

a | $0.5$ | $0.5$ | $0.2$ | $0.4$ |

b | $0.3$ | $0.2$ | $0.4$ | $0.6$ |

c | $0.3$ | $0.2$ | $0.4$ | $0.6$ |

∗ | 0 | 1 | 2 | 3 | 4 |
---|---|---|---|---|---|

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

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

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

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

4 | 4 | 4 | 4 | 3 | 0 |

X | ${\mathit{A}}_{\mathit{T}}\left(\mathit{x}\right)$ | ${\mathit{A}}_{\mathit{IT}}\left(\mathit{x}\right)$ | ${\mathit{A}}_{\mathit{IF}}\left(\mathit{x}\right)$ | ${\mathit{A}}_{\mathit{F}}\left(\mathit{x}\right)$ |
---|---|---|---|---|

0 | $0.7$ | $0.6$ | $0.1$ | $0.3$ |

1 | $0.5$ | $0.4$ | $0.2$ | $0.6$ |

2 | $0.3$ | $0.5$ | $0.4$ | $0.4$ |

3 | $0.3$ | $0.4$ | $0.4$ | $0.6$ |

4 | $0.3$ | $0.4$ | $0.4$ | $0.6$ |

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## Share and Cite

**MDPI and ACS Style**

Borzooei, R.A.; Zhang, X.; Smarandache, F.; Jun, Y.B.
Commutative Generalized Neutrosophic Ideals in *BCK*-Algebras. *Symmetry* **2018**, *10*, 350.
https://doi.org/10.3390/sym10080350

**AMA Style**

Borzooei RA, Zhang X, Smarandache F, Jun YB.
Commutative Generalized Neutrosophic Ideals in *BCK*-Algebras. *Symmetry*. 2018; 10(8):350.
https://doi.org/10.3390/sym10080350

**Chicago/Turabian Style**

Borzooei, Rajab Ali, Xiaohong Zhang, Florentin Smarandache, and Young Bae Jun.
2018. "Commutative Generalized Neutrosophic Ideals in *BCK*-Algebras" *Symmetry* 10, no. 8: 350.
https://doi.org/10.3390/sym10080350