# Q-Filters of Quantum B-Algebras and Basic Implication Algebras

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Proposition**

**1.**

**Proposition**

**2.**

**Definition**

**2.**

**Definition**

**3.**

- (1)
- x→y ≤ (y→z)↝(x→z), x↝y ≤ (y↝z)→(x↝z),
- (2)
- x ≤ (x↝y)→y, x ≤ (x→y)↝y;
- (3)
- x ≤ x;
- (4)
- x ≤ y, y ≤ x ⇒ x = y;
- (5)
- x ≤ y ⟺ x→y = 1 ⟺ x↝y = 1.

**Proposition**

**3.**

**Proposition**

**4.**

**Definition**

**4.**

**Proposition**

**5.**

**Proposition**

**6.**

**Definition**

**5.**

**Proposition**

**7.**

**Definition**

**6.**

**Definition**

**7.**

## 3. Q-Filters in Quantum B-Algebra

**Definition**

**9.**

- (1)
- F is an upper set of X, that is, F∈U(X);
- (2)
- for all x∈F, x→x∈F and x↝x∈F;
- (3)
- x∈F, y∈X, x→y∈F ⟹ y∈F.
- (4)
- A q-filter of X is normal if x→y∈F⟺ x↝y∈F.

**Proposition**

**8.**

**Proof.**

**Proposition**

**9.**

**Example**

**1.**

_{1}= {f}, F

_{2}= {a, b, f}; then, F

_{1}is a filter but not a q-filter of X, and F

_{2}is a normal q-filter of X.

**Theorem**

**1.**

_{F}on X as follows:

_{F}y ⟺ x→y∈F and y→x∈F, where x, y∈X.

_{F}is an equivalent relation on X;

_{F}is a congruence relation on X, that is, x≈

_{F}y ⟹ (z→x) ≈

_{F}(z→y), (x→z) ≈

_{F}(y→z), (z↝x) ≈

_{F}(z↝y), (x↝z) ≈

_{F}(y↝z), for all z∈X.

**Proof.**

_{F}x.

_{F}y, we can easily verify that y ≈

_{F}x.

_{F}y, y ≈

_{F}z. Then, x→y∈F, y→x∈F, y→z∈F, and z→y∈F, since

_{F}z.

_{F}is an equivalent relation on X.

_{F}y, then x→y∈F, y→x∈F. Since

_{F}(z→y).

**∈**F, (x→z)→(y→z)∈F. Thus, (x→z) ≈

_{F}(y→z).

_{F}y ⟹ (z↝x) ≈

_{F}(z↝y) and (x↝z) ≈

_{F}(y↝z). □

**Definition**

**10.**

_{F}= [a→a]

_{F}) ⟺ (there exists b∈X, such that [x↝y]

_{F}= [b↝b]

_{F}).

_{F}→, ↝, ≤) is a quantum B-algebra, where quotient operations → and ↝ are defined in a canonical way, and ≤ is defined as follows:

_{F}≤ [y]

_{F}⟺ (there exists a∈X such that [x]

_{F}→[y]

_{F}= [a→a]

_{F})

⟺ (there exists b∈X such that [x]

_{F}↝[y]

_{F}= [b↝b]

_{F}).

**Theorem**

**2.**

**Proof.**

_{F}= [a→a]

_{F}, then

_{F}= [a→a]

_{F}= [1]

_{F}.

_{F}= [1]

_{F}= [1↝1]

_{F}. Similarly, we can prove that the inverse is true. That is, Definition 10 (1) holds for A.

_{F}is defined as the following:

_{F}≤ [y]

_{F}⟺ [x]

_{F}→[y]

_{F}= [1]

_{F}.

_{F}.

_{F}≤ [x]

_{F}for any x∈A.

_{F}≤ [y]

_{F}and [y]

_{F}≤ [x]

_{F}, then [x]

_{F}→[y]

_{F}= [x→y]

_{F}= [1]

_{F}, [y]

_{F}→[x]

_{F}= [y→x]

_{F}= [1]

_{F}. By the definition of equivalent class, x→y = 1→(x→y) ∈F, y→x = 1→(y→x)∈F. It follows that x ≈

_{F}y; thus, [x]

_{F}= [y]

_{F}.

_{F}≤ [y]

_{F}and [y]

_{F}≤ [z]

_{F}, then [x]

_{F}→[y]

_{F}= [x→y]

_{F}= [1]

_{F}, [y]

_{F}→[z]

_{F}= [y→z]

_{F}= [1]

_{F}. Thus,

_{F}1, [x→z]

_{F}= [1]

_{F}. That is, [x]

_{F}→[z]

_{F}=[x→z]

_{F}= [1]

_{F}, [x]

_{F}≤ [z]

_{F}.

_{F}→, ↝, [1]

_{F}) is a quantum B-algebra and pseudo-BCI algebra. That is, Definition 10 (2) holds for A.

**Example**

**2.**

_{1}= {1}, F

_{2}= {a, b, 1}, F

_{3}= X, then F

_{i}(i = 1, 2, 3) are all normal q-filters of X, and quotient algebras (X/≈

_{Fi}→, ↝, [1]

_{Fi}) are pseudo-BCI algebras. Thus, X is a perfect quantum B-algebra.

**Example**

**3.**

_{F}→, ↝, ≤), (X/≈

_{X}→, ↝, ≤) are quantum B-algebras, and X is a perfect quantum B-algebra.

## 4. Basic Implication Algebras and Filters

**Definition**

**11.**

**Proposition**

**10.**

**Example**

**4.**

**Definition**

**12.**

**Proposition**

**11.**

**Proposition**

**12.**

**Proposition**

**13.**

**Example**

**5.**

**Example**

**6.**

**Definition**

**13.**

**Proposition**

**14.**

**Theorem**

**3.**

_{F}on X as follows:

_{F}y ⟺ x→y∈F and y→x∈F, where x, y∈X.

_{F}is a equivalent relation on X;

_{F}is a congruence relation on X, that is, x≈

_{F}y ⟹ (z→x) ≈

_{F}(z→y), (x→z) ≈

_{F}(y→z), for all z∈X.

**Proof**

_{F}x. Moreover, ∀x, y∈X, if x ≈

_{F}y, then y ≈

_{F}x.

_{F}y and y ≈

_{F}z. Then x→y∈F, y→x∈F, y→z∈F, and z→y∈F. Applying Definition 13 (4) and (5), we have

_{F}z.

_{F}is a equivalent relation on X.

_{F}y. By the definition of bianary relation ≈

_{F}, we have x→y∈F, y→x∈F. Using Definition 13 (4),

_{F}(z→y). Moreover, using Definition 13 (5), we have

_{F}(y→z). □

**Theorem**

**4.**

_{F}as follows:

_{F}→[y]

_{F}= [x]

_{F}→[y]

_{F}, ∀x, y∈X;

[x]

_{F}≤ [y]

_{F}⟺ [x]

_{F}→[y]

_{F}= [e]

_{F}, ∀x, y∈X.

_{F}, →, ≤, [e]

_{F}) is a normal basic implication algebra, and (X, →, ≤, e) ∼ (X/≈

_{F}, →, ≤, [e]

_{F}).

**Proof.**

_{F}is a partial order.

_{F}≤ [x]

_{F}.

_{F}≤ [y]

_{F}and [y]

_{F}≤ [x]

_{F}, then

_{F}→[y]

_{F}= [x→y]

_{F}= [e]

_{F}, [y]

_{F}→[x]

_{F}= [y→x]

_{F}= [e]

_{F}.

_{F}= [y]

_{F}.

_{F}≤ [y]

_{F}and [y]

_{F}≤ [z]

_{F}, then

_{F}→[y]

_{F}= [x→y]

_{F}= [e]

_{F}, [y]

_{F}→[z]

_{F}= [y→z]

_{F}= [e]

_{F}.

_{F}, we have

_{F}e, that is, [x→z]

_{F}= [e]

_{F}. This means that [x]

_{F}≤ [z]

_{F}. It follows that the binary relation ≤ on X/≈

_{F}is a partially order.

_{F}→, ≤, [e]

_{F}) is a normal basic implication algebra, and (X, →, ≤, e) ∼ (X/≈

_{F}→, ≤, [e]

_{F}) in the homomorphism mapping f: X→X/≈

_{F}; f(x) = [x]

_{F}. □

**Example**

**7.**

_{F}, →, [1]

_{F}).

**Example**

**8.**

_{F}, →, [1]

_{F}) is presented as the following Table 11, where X/≈

_{F}= {{a}, {b, c}, [1]

_{F}= {1, d}}. Moreover, (X, →, ≤, 1) ∼ (X/≈

_{F}→, [1]

_{F}).

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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$\mathit{\to}$ | a | b | c | d | e | f |
---|---|---|---|---|---|---|

a | f | a | c | c | c | f |

b | f | a | c | c | c | f |

c | c | c | f | a | b | c |

d | c | c | f | f | a | c |

e | c | c | f | f | a | c |

f | a | b | c | d | e | f |

↝ | a | b | c | d | e | f |
---|---|---|---|---|---|---|

a | f | a | c | c | d | f |

b | f | f | c | c | c | f |

c | c | c | f | a | a | c |

d | c | c | f | a | a | c |

e | c | c | f | f | a | c |

f | a | b | c | d | e | f |

$\mathit{\to}$ | a | b | c | d | e | 1 |
---|---|---|---|---|---|---|

a | 1 | a | c | c | c | 1 |

b | 1 | 1 | c | c | c | 1 |

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

d | c | c | 1 | 1 | a | c |

e | c | c | 1 | 1 | 1 | c |

1 | a | b | c | d | e | 1 |

↝ | a | b | c | d | e | 1 |
---|---|---|---|---|---|---|

a | 1 | a | c | c | d | 1 |

b | 1 | 1 | c | c | c | 1 |

c | c | c | 1 | a | a | c |

d | c | c | 1 | 1 | a | c |

e | c | c | 1 | 1 | 1 | c |

1 | a | b | c | d | e | 1 |

$\mathit{\to}$ | a | b | c | d | e | f |
---|---|---|---|---|---|---|

a | f | a | c | c | c | f |

b | f | f | c | c | c | f |

c | c | c | f | a | b | c |

d | c | c | f | f | a | c |

e | c | c | f | f | f | c |

f | a | b | c | d | e | f |

↝ | a | b | c | d | e | f |
---|---|---|---|---|---|---|

a | f | a | c | c | d | f |

b | f | f | c | c | c | f |

c | c | c | f | a | a | c |

d | c | c | f | f | a | c |

e | c | c | f | f | a | c |

f | a | b | c | d | e | f |

$\mathit{\to}$ | a | b | c | d | 1 |
---|---|---|---|---|---|

a | 1 | 1 | 1 | 1 | 1 |

b | d | 1 | 1 | 1 | 1 |

c | d | d | 1 | 1 | 1 |

d | b | c | d | 1 | 1 |

1 | b | b | c | b | 1 |

$\mathit{\to}$ | a | b | c | d | 1 |
---|---|---|---|---|---|

a | 1 | 1 | 1 | 1 | 1 |

b | c | 1 | 1 | 1 | 1 |

c | c | c | 1 | 1 | 1 |

d | a | c | a | 1 | c |

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

$\mathit{\to}$ | a | b | c | d | 1 |
---|---|---|---|---|---|

a | 1 | 1 | 1 | 1 | 1 |

b | d | 1 | 1 | 1 | 1 |

c | b | d | 1 | d | 1 |

d | a | c | c | 1 | 1 |

1 | a | b | c | d | 1 |

$\mathit{\to}$ | a | b | c | d | 1 |
---|---|---|---|---|---|

a | 1 | 1 | 1 | 1 | 1 |

b | d | 1 | 1 | 1 | 1 |

c | b | d | 1 | d | 1 |

d | a | b | c | 1 | 1 |

1 | a | b | c | d | 1 |

$\mathit{\to}$ | {a} | {b,c} | [1]_{F} |
---|---|---|---|

{a} | [1]_{F} | [1]_{F} | [1]_{F} |

{b,c} | {b,c} | [1]_{F} | [1]_{F} |

[1]_{F} | {a} | {b,c} | [1]_{F} |

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## Share and Cite

**MDPI and ACS Style**

Zhang, X.; Borzooei, R.A.; Jun, Y.B.
Q-Filters of Quantum B-Algebras and Basic Implication Algebras. *Symmetry* **2018**, *10*, 573.
https://doi.org/10.3390/sym10110573

**AMA Style**

Zhang X, Borzooei RA, Jun YB.
Q-Filters of Quantum B-Algebras and Basic Implication Algebras. *Symmetry*. 2018; 10(11):573.
https://doi.org/10.3390/sym10110573

**Chicago/Turabian Style**

Zhang, Xiaohong, Rajab Ali Borzooei, and Young Bae Jun.
2018. "Q-Filters of Quantum B-Algebras and Basic Implication Algebras" *Symmetry* 10, no. 11: 573.
https://doi.org/10.3390/sym10110573