# On Two Conjectures of Abel Grassmann’s Groupoids

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- (i)
- x = xy and y
^{2}= yx imply that x = y, - (ii)
- x = yx and y
^{2}= xy imply that x = y.

**Conjecture****1**.- Conditions (i) and (ii) above are equivalent for AG-groupoids.
**Conjecture****2**.- Every AG-3-band is quasi-cancellative.

## 2. Preliminaries

**Definition**

**1.**

**Proposition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

^{2}, we say that a is a 4-potent. The AG-groupoid S is an AG-4-band (or a 4-band) if all of its elements are 4-potents.

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

- (i)
- a = a∗b and b
^{2}= b∗a imply a = b; - (ii)
- a = b∗a and b
^{2}= a∗b imply a = b.

## 3. Quasi-Cancellativity of AG-3-Bands and AG-(4, 1)-Bands

**Lemma**

**1.**

^{2}= y∗x, then

- (1)
- x
^{2}= x^{2}∗y^{2}= x∗x^{2}; - (2)
- y
^{2}∗y^{2}= y^{2}∗x^{2}= (x^{2}∗x)∗y; - (3)
- y
^{2}∗y^{2}= y^{2}∗x = x; - (4)
- x
^{2}∗y = y^{2}∗x.

**Proof.**

^{2}= y∗x, applying Definition 1 and Proposition 1 we have

^{2}= (x∗y)∗(x∗y) = (x∗x)∗(y∗y) = x

^{2}∗y

^{2};

^{2}∗y

^{2}= (x∗x)∗(y∗x) = (x∗y)∗(x∗x) = x∗x

^{2}.

^{2}∗y

^{2}= (y∗x)∗(y∗x) = (y∗y)∗(x∗x) = y

^{2}∗x

^{2};

^{2}∗x

^{2}= (y∗x)∗x

^{2}= (x

^{2}∗x)∗y.

^{2}∗y

^{2}= (y∗x)∗ y

^{2}= (y∗x)∗(y∗y) = y

^{2}∗(x∗y) = y

^{2}∗x;

^{2}∗x = (y∗y)∗x = (x∗y)∗y = x∗y = x.

^{2}∗y = (x∗x)∗y = (y∗x)∗x = y

^{2}∗x.

**Lemma**

**2.**

^{2}= x∗y, then

- (1)
- x
^{2}= y^{2}∗x^{2}= x^{2}∗x; - (2)
- y
^{2}∗y^{2}= x^{2}∗y^{2}= (y^{2}∗y)∗x^{2}; - (3)
- y
^{2}∗x^{2}= x^{2}∗y^{2}= x^{2}; - (4)
- x∗y
^{2}= y^{2}∗y^{2}, x^{2}∗y = x^{2}.

**Proof.**

^{2}= x∗y, applying Definition 1 and Proposition 1 we have

^{2}= (y∗x)∗(y∗x) = (y∗y)∗(x∗x) = y

^{2}∗x

^{2};

^{2}∗x

^{2}= (x∗y)∗x

^{2}= (x∗y)∗(x∗x) = x

^{2}∗(y∗x) = x

^{2}∗x.

^{2}∗y

^{2}= (x∗y)∗(x∗y) = (x∗x)∗(y∗y) = x

^{2}∗y

^{2};

^{2}∗y

^{2}= (x

^{2}∗x)∗y

^{2}= (x

^{2}∗x)∗(y∗y) = (x

^{2}∗y)∗(x∗y) = (x

^{2}∗y)∗y

^{2}= (y

^{2}∗y)∗x

^{2}.

^{2}∗x

^{2}= (y∗y)∗(x∗x) = (y∗x)∗(y∗x) = x∗x = x

^{2};

^{2}∗y

^{2}= (y

^{2}∗y)∗x

^{2}= (y

^{2}∗y)∗(x

^{2}∗x) = (y

^{2}∗x

^{2})∗(y∗x) = (y

^{2}∗x

^{2})∗x = x

^{2}∗x = x

^{2}.

^{2}= (y∗x)∗y

^{2}= (y∗x)∗(y∗y) = (y∗y)∗(x∗y) = y

^{2}∗y

^{2};

^{2}∗y = (x∗x)∗y = (y∗x)∗x = x∗x = x

^{2}.

**Theorem**

**1.**

**Proof.**

^{2}= y∗x) ⇒ x = y.

^{2}= x, y

^{2}∗y = y, y∗y

^{2}= y.

^{2}= x∗x

^{2}= x. Using Lemma 1 (2) and (3) we have

^{2}∗y

^{2}= (y∗y)∗ y

^{2}= (y

^{2}∗y) = y∗ y = y

^{2}.

^{2}= y∗x = y∗y

^{2}= y.

^{2}= x∗y) ⇒ x = y.

^{2}∗x = x, y

^{2}∗y = y.

^{2}= x

^{2}∗x = x. From this, using Lemma 2 (4) we get that

^{2}= x

^{2}∗y = x∗ y = y

^{2}.

^{2}= x∗y = y

^{2}∗y = y.

**Definition**

**8.**

^{2}∗a

^{2}= a.

**Example**

**1.**

^{2}∗7

^{2}= 2 ≠ 7.

**Example**

**2.**

**Proposition**

**4.**

**Theorem**

**2.**

**Proof.**

^{2}∗a

^{2}= a for any a∈S.

^{2}= y∗x) ⇒ x = y.

^{2}∗y

^{2}= y. Using Lemma 1 (3), we get that x = y

^{2}∗y

^{2}= y.

^{2}= x∗y) ⇒ x = y.

^{2}∗y

^{2}= y. From this, using Lemma 2 (2), (3) and (4) we have

^{2}= x

^{2}∗y = x

^{2}∗y

^{2}= y

^{2}∗y

^{2}= y.

^{2}= x

^{2}∗x = y∗x = x.

^{2}= y. □

^{2}= 2 = 7∗2, but 7 ≠ 2;

^{2}= 2 = 2∗7, but 2 ≠ 7.

**Example**

**3.**

## 4. Left (Right) Quasi-Cancellative and Power-Cancellative AG-Groupoids

**Definition**

**9.**

- (i)
- a = a∗b and b
^{2}= b∗a imply a = b;S is called a right quasi-cancellative AG-groupoid, if for any a, b$\in $S, - (ii)
- a = b∗a and b
^{2}= a∗b imply a = b.

**Theorem**

**3.**

**Proof.**

^{2}= x∗y, x, y∈S.

^{2}= x

^{2}∗x. And, by Definition 5, x

^{2}= x

^{2}∗x = (x∗x)∗ x = x∗(x∗x) = x∗x

^{2}. Then

^{2}= x

^{2}∗x and x

^{2}= x∗x

^{2}.

^{2}= x. From this, using Lemma 2 (4) we have

^{2}= x

^{2}∗y = x∗y.

^{2}= x∗y = y∗x.

^{2}= y∗x, x, y∈S.

^{2}= x∗x

^{2}. And, from this and Definition 5, x

^{2}∗x = (x∗x)∗x = x∗(x∗x) = x∗x

^{2}= x

^{2}. Then

^{2}= x∗x

^{2}and x

^{2}= x

^{2}∗x.

^{2}= x. Moreover, from y

^{2}= y∗x, by Definition 5 we have

^{2}∗x = (y∗x)∗ x = x∗ (y∗x) = x∗y

^{2}.

^{2}∗y

^{2}= y

^{2}∗x = x∗y

^{2}.

^{2}∗x (Lemma 1 (3)), and y

^{2}∗y

^{2}= x∗y

^{2}.

^{2}from the right quasi-cancellative law. Hence,

^{2}= y∗x and y

^{2}= x = x∗y.

**Theorem**

**4.**

**Proof.**

^{2}= x∗y, x, y∈S.

^{2}= y

^{2}∗x

^{2}and y

^{2}∗y

^{2}= x

^{2}∗y

^{2}.

^{2}= y

^{2}. Moreover, using Lemma 2 (1) and (4) we can get that

^{2}= (y∗x)∗ x

^{2}= (x

^{2}∗x)∗y = x

^{2}∗y = x

^{2}.

^{2}= x

^{2}∗x and x

^{2}= x∗x

^{2}.

^{2}. Hence, x∗y = y

^{2}= x

^{2}= x = y∗x and

^{2}= y∗x.

^{2}= y∗x, x, y∈S.

^{2}∗x

^{2}= x

^{2}∗ (x∗x) = x∗(x

^{2}∗x) = (y

^{2}∗x)∗(x

^{2}∗x) = (y

^{2}∗x

^{2})∗x

^{2}= (y

^{2}∗y

^{2})∗x

^{2}= x∗x

^{2}= x

^{2}.

^{2}∗x = x

^{2}∗ (x∗y) = x∗(x

^{2}∗y) = x∗(y

^{2}∗x) = y

^{2}∗(x∗ x) = y

^{2}∗x

^{2}= x.

^{2}∗x and x

^{2}∗x

^{2}= x∗x

^{2}.

^{2}. Hence,

^{2}= x

^{2}∗y

^{2}= x∗y

^{2}= x∗(y∗y) = y∗(x∗y) = y∗x = y

^{2}.

^{2}= x∗y.

**Theorem**

**5.**

**Definition**

**10.**

^{2}= b

^{2}implies a = b.

**Example**

**4.**

**Definition**

**11.**

^{2}∗(a

^{2}∗a

^{2}) = (a

^{2}∗a

^{2})∗ a

^{2}.

**Example**

**5.**

^{2}= 4∗4 = 3, but 4

^{2}∗(4

^{2}∗4

^{2}) = 4 ≠ 3 = (4

^{2}∗4

^{2})∗4

^{2}.

**Theorem**

**6.**

**Proof.**

^{2}= y∗x, x, y∈S.

^{2}= x∗ x

^{2}= (y

^{2}∗x

^{2})∗x

^{2}= (y

^{2}∗x

^{2})∗(x∗x) = (y

^{2}∗x)∗(x

^{2}∗x) = x∗(x

^{2}∗x).

^{2}∗x

^{2}= (x∗(x

^{2}∗x))∗ x

^{2}= (x∗(x

^{2}∗x))∗(x∗x) = x

^{2}∗((x

^{2}∗x)∗x) = x

^{2}∗((x∗x)∗x

^{2}) = x

^{2}∗(x

^{2}∗x

^{2}).

^{2}∗(x

^{2}∗x

^{2}) = (x

^{2}∗x

^{2})∗x

^{2}. It follows that

^{2}∗x

^{2}= x

^{2}∗(x

^{2}∗x

^{2}) and (x

^{2})

^{2}= x

^{2}∗x

^{2}= (x

^{2}∗x

^{2})∗x

^{2}.

^{2}∗x

^{2}= x

^{2}. Hence,

^{2})

^{2}= x

^{2}∗x

^{2}= (x

^{2}∗x

^{2})∗ x

^{2}= (x

^{2}∗x)∗(x

^{2}∗x) = (x

^{2}∗x)

^{2}.

^{2}= x

^{2}∗x. Moreover, using Lemma 1 again, we have

^{2}∗x)

^{2}= (y

^{2}∗x )∗x = x

^{2}∗y

^{2}= x

^{2};

^{2})

^{2}= (x∗y

^{2})∗(x∗y

^{2}) = x

^{2}∗(y

^{2}∗y

^{2}) = x

^{2}∗x = x

^{2}.

^{2}∗x)

^{2}= (x∗y

^{2})

^{2}. Using Definition 10, (y

^{2}∗x) = (x∗y

^{2}). Hence,

^{2}∗x (Lemma 1 (3)), and y

^{2}∗y

^{2}= y

^{2}∗x = x∗y

^{2}.

^{2}. Therefore,

^{2}= y∗x, and y

^{2}∗y

^{2}= x = x∗y.

## 5. Conclusions

- (1)
- Every AG-3-band is quasi-cancellative;
- (2)
- Every AG-(4,1)-band is quasi-cancellative;
- (3)
- For an AG*-groupoid, or an AG**-groupoid, or a power-cancellative and locally power- associative AG-groupoid, it is left quasi-cancellative if and only if it is right quasi-cancellative;
- (4)
- Every left quasi-cancellative AG-groupoid is right quasi-cancellative.

## Author Contributions

## Funding

## Conflicts of Interest

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∗ | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|

1 | 1 | 2 | 2 | 5 | 6 | 4 | 7 |

2 | 2 | 2 | 2 | 5 | 6 | 4 | 2 |

3 | 2 | 2 | 3 | 5 | 6 | 4 | 2 |

4 | 6 | 6 | 6 | 4 | 2 | 5 | 6 |

5 | 4 | 4 | 4 | 6 | 5 | 2 | 4 |

6 | 5 | 5 | 5 | 2 | 4 | 6 | 5 |

7 | 7 | 2 | 2 | 5 | 6 | 4 | 2 |

∗ | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

1 | 1 | 1 | 1 | 1 | 1 |

2 | 1 | 2 | 2 | 4 | 5 |

3 | 1 | 2 | 3 | 4 | 5 |

4 | 1 | 4 | 4 | 5 | 2 |

5 | 1 | 5 | 5 | 2 | 4 |

∗ | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|

1 | 1 | 2 | 2 | 5 | 6 | 4 | 2 |

2 | 2 | 2 | 2 | 5 | 6 | 4 | 2 |

3 | 2 | 2 | 3 | 5 | 6 | 4 | 7 |

4 | 6 | 6 | 6 | 4 | 2 | 5 | 6 |

5 | 4 | 4 | 4 | 6 | 5 | 2 | 4 |

6 | 5 | 5 | 5 | 2 | 4 | 6 | 5 |

7 | 2 | 2 | 7 | 5 | 6 | 4 | 3 |

∗ | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

1 | 1 | 1 | 1 | 1 | 1 |

2 | 1 | 2 | 2 | 4 | 5 |

3 | 1 | 2 | 3 | 4 | 5 |

4 | 1 | 5 | 5 | 2 | 4 |

5 | 1 | 4 | 4 | 5 | 2 |

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

1 | 2 | 1 | 3 | 4 |

2 | 3 | 4 | 2 | 1 |

3 | 4 | 3 | 1 | 2 |

4 | 1 | 2 | 4 | 3 |

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**MDPI and ACS Style**

Zhang, X.; Ma, Y.; Yu, P.
On Two Conjectures of Abel Grassmann’s Groupoids. *Symmetry* **2019**, *11*, 816.
https://doi.org/10.3390/sym11060816

**AMA Style**

Zhang X, Ma Y, Yu P.
On Two Conjectures of Abel Grassmann’s Groupoids. *Symmetry*. 2019; 11(6):816.
https://doi.org/10.3390/sym11060816

**Chicago/Turabian Style**

Zhang, Xiaohong, Yingcang Ma, and Peng Yu.
2019. "On Two Conjectures of Abel Grassmann’s Groupoids" *Symmetry* 11, no. 6: 816.
https://doi.org/10.3390/sym11060816