# Length-Fuzzy Subalgebras in BCK/BCI-Algebras

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

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

**Definition**

**1.**

- fuzzy subalgebra of $(X,\ast ,0)$ with type 1 (briefly, 1-fuzzy subalgebra of $(X,\ast ,0)$) if$$\begin{array}{c}\hfill (\forall x,y\in X)\left(\mu (x\ast y)\ge min\left\{\mu \right(x),\mu (y\left)\right\}\right),\end{array}$$
- fuzzy subalgebra of $(X,\ast ,0)$ with type 2 (briefly, 2-fuzzy subalgebra of $(X,\ast ,0)$) if$$\begin{array}{c}\hfill (\forall x,y\in X)\left(\mu (x\ast y)\le min\left\{\mu \right(x),\mu (y\left)\right\}\right),\end{array}$$
- fuzzy subalgebra of $(X,\ast ,0)$ with type 3 (briefly, 3-fuzzy subalgebra of $(X,\ast ,0)$) if$$\begin{array}{c}\hfill (\forall x,y\in X)\left(\mu (x\ast y)\ge max\left\{\mu \right(x),\mu (y\left)\right\}\right),\end{array}$$
- fuzzy subalgebra of $(X,\ast ,0)$ with type 4 (briefly, 4-fuzzy subalgebra of $(X,\ast ,0)$) if$$\begin{array}{c}\hfill (\forall x,y\in X)\left(\mu (x\ast y)\le max\left\{\mu \right(x),\mu (y\left)\right\}\right).\end{array}$$

**Definition**

**2.**

## 3. Length-Fuzzy Subalgebras

**Definition**

**3.**

**Definition**

**4.**

**Example**

**1.**

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Example**

**2.**

**Theorem**

**2.**

**Proof.**

**Corollary**

**1.**

**Example**

**3.**

**Theorem**

**3.**

**Proof.**

**Corollary**

**2.**

**Example**

**4.**

**Theorem**

**4.**

**Proof.**

**Corollary**

**3.**

**Corollary**

**4.**

**Example**

**5.**

**Theorem**

**5.**

**Proof.**

**Corollary**

**5.**

**Corollary**

**6.**

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

**Proof.**

**Corollary**

**7.**

**Theorem**

**8.**

**Proof.**

**Corollary**

**8.**

**Corollary**

**9.**

**Theorem**

**9.**

**Proof.**

**Corollary**

**10.**

**Theorem**

**10.**

**Proof.**

**Corollary**

**11.**

**Theorem**

**11.**

**Proof.**

**Corollary**

**12.**

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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∗ | 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 | 3 | 4 | 1 | 0 |

X | 0 | 1 | 2 | 3 | 4 |
---|---|---|---|---|---|

${\tilde{\mu}}_{\ell}$ | $0.6$ | $0.4$ | $0.5$ | $0.3$ | $0.2$ |

∗ | 0 | 1 | 2 | a | b |
---|---|---|---|---|---|

0 | 0 | 0 | 0 | a | a |

1 | 1 | 0 | 1 | b | a |

2 | 2 | 2 | 0 | a | a |

a | a | a | a | 0 | 0 |

b | b | a | b | 1 | 0 |

X | 0 | 1 | 2 | a | b |
---|---|---|---|---|---|

${\tilde{\mu}}_{\ell}$ | $0.6$ | $0.2$ | $0.5$ | $0.3$ | $0.2$ |

X | 0 | 1 | 2 | a | b |
---|---|---|---|---|---|

${\tilde{\mu}}_{\ell}$ | $0.2$ | $0.4$ | $0.3$ | $0.6$ | $0.6$ |

∗ | 0 | 1 | a | b | c |
---|---|---|---|---|---|

0 | 0 | 0 | a | b | c |

1 | 1 | 0 | a | b | c |

a | a | a | 0 | c | b |

b | b | b | c | 0 | a |

c | c | c | b | a | 0 |

X | 0 | 1 | a | b | c |
---|---|---|---|---|---|

${\tilde{\mu}}_{\ell}$ | $0.8$ | $0.7$ | $0.5$ | $0.3$ | $0.3$ |

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## Share and Cite

**MDPI and ACS Style**

Jun, Y.B.; Song, S.-Z.; Kim, S.J. Length-Fuzzy Subalgebras in *BCK*/*BCI*-Algebras. *Mathematics* **2018**, *6*, 11.
https://doi.org/10.3390/math6010011

**AMA Style**

Jun YB, Song S-Z, Kim SJ. Length-Fuzzy Subalgebras in *BCK*/*BCI*-Algebras. *Mathematics*. 2018; 6(1):11.
https://doi.org/10.3390/math6010011

**Chicago/Turabian Style**

Jun, Young Bae, Seok-Zun Song, and Seon Jeong Kim. 2018. "Length-Fuzzy Subalgebras in *BCK*/*BCI*-Algebras" *Mathematics* 6, no. 1: 11.
https://doi.org/10.3390/math6010011