# Some Results on Various Cancellative CA-Groupoids and Variant CA-Groupoids

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Proposition**

**1.**

**Proposition**

**2.**

**Proposition**

**3.**

**Proposition**

**4.**

^{2}= a, then S is commutative (thus, S is a commutative semigroup).

**Proposition**

**5.**

_{1}, S

_{2}be two CA-groupoids. Then the direct product S

_{1}× S

_{2}is a CA-groupoid.

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

^{2}= xy and y

^{2}= yx imply x = y;

^{2}= yx and y

^{2}= xy imply x = y.

^{2}= xy = y

^{2}imply x = y.

## 3. Cancellation Properties of CA-Groupoids

**Definition**

**5.**

**Example**

**1.**

**Definition**

**6.**

**Example**

**2.**

**Theorem**

**1.**

- (1)
- if a is left cancellative, then a is right cancellative, thus a is cancellative;
- (2)
- if a and b are left cancellative, then ab is right cancellative;
- (3)
- if a is right cancellative and b is left cancellative, then ab is right cancellative;
- (4)
- if ab is right cancellative, then ab = ba;
- (5)
- if ab is cancellative, then b is cancellative;
- (6)
- if ab is cancellative, then a and b are cancellative;
- (7)
- if a and ab are right cancellative, and b is left cancellative, then a is cancellative;
- (8)
- if a and ab are right cancellative, and b is left cancellative, then ab is cancellative.

**Proof.**

**Corollary**

**1.**

- (1)
- S is a left cancellative CA-groupoid;
- (2)
- S is a right cancellative CA-groupoid;
- (3)
- S is a cancellative and commutative semigroup;
- (4)
- S is a cancellative CA-groupoid.

**Proof.**

**Corollary**

**2.**

**Proof.**

**Corollary**

**3.**

**Proof.**

**Example**

**3.**

**Open Problem 1**(to prove or give a counterexample): Is any weak cancellative CA-groupoid necessarily cancellative?

**Theorem**

**2.**

**Proof.**

**Example**

**4.**

**Theorem**

**3.**

_{1}, S

_{2}are CA-groupoids, then the direct product S

_{1}×S

_{2}of S

_{1}and S

_{2}is a CA-groupoid. If a$\in $S

_{1}, b$\in $S

_{2}, a and b are cancellative, then(a, b)$\in $S

_{1}× S

_{2}is cancellative.

**Proof.**

_{1}and S

_{2}are CA-groupoids. By Proposition 5, S

_{1}× S

_{2}is a CA-groupoid. Let a$\in $S

_{1}, b$\in $S

_{2}, a and b be cancellative. For any$\left({x}_{1},{x}_{2}\right),\left({y}_{1},{y}_{2}\right)\in {S}_{1}\times {S}_{2}$, if $\left(a,b\right)\ast \left({x}_{1},{x}_{2}\right)=\left(a,b\right)\ast \left({y}_{1},{y}_{2}\right)$, then:

_{1}, bx

_{2}) = (ay

_{1}, by

_{2})

_{1}= ay

_{1}, bx

_{2}= by

_{2}

_{1}= y

_{1}, x

_{2}= y

_{2}. (since a and b are cancellative)

_{1}, x

_{2}) = (y

_{1}, y

_{2}).

## 4. Separability and Quasi-Cancellability of CA-Groupoids

**Definition**

**7.**

^{2}= xy and y

^{2}= yx (x

^{2}= yx and y

^{2}= xy) imply x = y. (2) S is called to be separative, if it is both left and right separative. (3) S is called to be quasi-separative, if for all x, y$\in $S, x

^{2}= xy = y

^{2}implies x = y.

**Example**

**5.**

**Example**

**6.**

**Theorem**

**4.**

- (1)
- S is separative;
- (2)
- S is left separative;
- (3)
- S is right separative;
- (4)
- S is quasi-separative.

**Proof.**

^{2}= yx and y

^{2}= xy, then (by Proposition 1 (1)):

^{2}= (xy)(xy) = (xy) y

^{2}= (xy)(yy) = (yx)(yy) = x

^{2}(yy) = (xx)(yy) = (yx)(yx) = (xy)(yx);

^{2}= (yx)(yx) = (xy)(yx) = (xx)(yy) = x

^{2}y

^{2}= (yx)(xy).

^{2}= yx and y

^{2}= xy, we get that x

^{2}= xy and y

^{2}= yx. Applying the condition that S is left separative, by Definition 7 again, we have x = y. This means that S is right separative.

^{2}= xy = y

^{2}, then (by Proposition 1 (1)):

^{2}= (xy)(xy) = x

^{2}(xy) = (xx)(xy) = (yx)(xx) = (yx)x

^{2}= (yx)(xy);

^{2}= (yx)(yx) = (xy)(yx).

^{2}= xy = y

^{2}, we get that x

^{2}= yx and y

^{2}= xy. Applying the condition that S is right separative, by Definition 7 again, we have x = y. This means that S is quasi-separative.

^{2}= (xy)(xy) = (yx)(xy) = (yy)(xx) = y

^{2}x

^{2};

^{2}= (yx)(yx) = (xy)(yx) = (xx)(yy) = x

^{2}y

^{2}.

^{2}= [(yx)(xy)] [(yx)(xy)] = [(yx)(xy)] [(yy)(xx)] = [(yx)(xy)] (y

^{2}x

^{2}) = [(yx)(xy)] (xy)

^{2}=

^{2}y

^{2}) [(xy)(xy)] =

^{2}[(xy)(xy)] = [(yx)(yx)] [(xy)(xy)] = [(xy)(yx)] [(xy)(yx)] = [(xy)(yx)]

^{2}.

^{2}= xy and y

^{2}= yx, then

^{4})

^{2}= (y

^{2}y

^{2})

^{2}= [(yx)(yx)]

^{2}= [(xy)(yx)]

^{2}= [(yx)(xy)]

^{2}= (y

^{2}x

^{2})

^{2}= (y

^{2}x

^{2})(y

^{2}x

^{2}) = (x

^{2}y

^{2})(y

^{2}x

^{2}) = (x

^{2}x

^{2})(y

^{2}y

^{2}) =x

^{4}y

^{4};

^{4})

^{2}= (x

^{2}x

^{2})

^{2}= [(xy)(xy)]

^{2}= [(yx)(xy)]

^{2}= [(yy)(xx)]

^{2}= (y

^{2}x

^{2})

^{2}= (y

^{2}x

^{2})(y

^{2}x

^{2}) = (x

^{2}y

^{2})(y

^{2}x

^{2}) = (x

^{2}x

^{2})(y

^{2}y

^{2}) =x

^{4}y

^{4}.

^{4}=y

^{4}. Thus,

^{2}= (x

^{2})

^{2}= x

^{4}= y

^{4}= (y

^{2})

^{2}= (yx)

^{2}= (yx)(yx) = (xy)(yx).

^{2}= (xy)(yx) = (yx)

^{2}. Since S is quasi-separative, by Definition 7 we have xy = yx. From this, using x

^{2}= xy and y

^{2}= yx, we have x

^{2}= xy = y

^{2}. Applying the condition that S is quasi-separative, by Definition 7 again, we have x = y. This means that S is left separative. □

**Proposition**

**6.**

**Proof.**

^{2}= xy = y

^{2}, then xx = xy and xy = yy. Using cancellability of S, we have x = y. This means that S is separative.

**Example**

**7.**

**Definition**

**8.**

**Definition**

**9.**

^{2}= yx (x = yx and y

^{2}= xy) imply x = y. S is called quasi-cancellative, if it is both left and right quasi- cancellative.

**Example**

**8.**

**Theorem**

**5.**

**Proof.**

^{2}= xy, then (by Proposition 1 (1)):

^{2}= (yx)(yx) = (xy)(yx) = (xx)(yy) = x

^{2}y

^{2};

^{2})

^{2}= y

^{2}y

^{2}= (xy)(xy) = (yx)(xy) = (yy)(xx) = y

^{2}x

^{2}.

^{2}= y

^{2}. Thus:

^{2}= x

^{2}= (yx)(yx) = (xy)(yx);

^{2}= (yx)(yx) = x(yx) = x(xy) = (yx)(xy).

^{2}= xy, we have x = xy and y

^{2}= yx, applying the definition of left quasi-cancellative, we get that x = y. Therefore, S is right quasi-cancellative. □

**Open Problem 2**(to prove or give a counterexample): Is any right quasi-cancellative CA-groupoid necessarily left quasi-cancellative?

**Theorem**

**6.**

- (1)
- Every CA-band is quasi-cancellative.
- (2)
- Every CA-3-band is quasi-cancellative.
- (3)
- Everyquasi-separativeCA-groupoid is quasi-cancellative;
- (4)
- Every separative(or left-, right-separative) CA-groupoid is quasi-cancellative.

**Proof.**

^{2}= yx, then (by Definition 8) x = x

^{2}, y = y

^{2}. It follows that:

^{2}= (xy)(xy) = (yx)(xy) = y

^{2}(xy) = y(xy) = yx = y

^{2}= y.

^{2}= yx, then (by Definition 8) x = xx

^{2}= x

^{2}x, y = yy

^{2}= y

^{2}y. Furthermore:

^{2}= yx = y(xy) = y(yx) = yy

^{2}= y,

^{2}) = y

^{2}(xy) = y

^{2}x = yx = y

^{2}= y.

^{2}= yx, then:

^{2}= xx = x(xy) = y(xx) = x(yx) = xy

^{2}= x(yy) = y(xy) = yx = y

^{2}

^{2}= yx = x

^{2}. By Definition 7 we have x = y. This means that S is left quasi-cancellative. Applying Theorem 5, we get that S is right quasi-cancellative. Hence, S is quasi-cancellative.

**Example**

**9.**

**Definition**

**10.**

^{2}= y

^{2}implies x = y.

**Example**

**10.**

**Example**

**11.**

^{2}= 2

^{2}= 1, but 1$\text{}\ne \text{}$2.

**Theorem**

**7.**

- (1)
- S is commutative, and S is a commutative semigroup.
- (2)
- S is separative.

**Proof.**

^{2}= (xy)(xy) = (yx)(xy) = (yy)(xx) = y

^{2}x

^{2};

^{2}= (yx)(yx) = (xy)(yx) = (xx)(yy) = x

^{2}y

^{2}.

^{2}= [(yx)(xy)] [(yx)(xy)] = [(yx)(xy)] [(yy)(xx)] = [(yx)(xy)] (y

^{2}x

^{2}) = [(yx)(xy)] (xy)

^{2}=

^{2}y

^{2}) [(xy)(xy)] =

^{2}[(xy)(xy)] = [(yx)(yx)] [(xy)(xy)] = [(xy)(yx)] [(xy)(yx)] = [(xy)(yx)]

^{2}

^{2}= y

^{2}x

^{2}= (yx)(xy) = (xy)(yx) = x

^{2}y

^{2}= (yx)

^{2}.

^{2}= xy = y

^{2}, then (by Definition 10), x = y. This means that S is quasi-separative. Applying Theorem 4, we know that S is separative. □

## 5. Variant CA-Groupoids

**Definition**

**11.**

^{2}$\ne $e. Where, e is called a quasi-right unite element of S.

**Example**

**12.**

**Example**

**13.**

**Example**

**14.**

_{1}=$\left\{\left(\begin{array}{cc}a& 0\\ 0& 0\end{array}\right):\text{\hspace{0.17em}}a\text{}is\text{}a\text{}integral\text{}number\right\}$, S

_{2}=$\left\{\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right),\text{\hspace{0.17em}}\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)\right\}$,

- (1)
- if x, y∈ S
_{1}, x + y is common matrix addition; - (2)
- if x∈S
_{1}and y∈S_{2}, x + y =$\left(\begin{array}{cc}a+1& 0\\ 0& 0\end{array}\right)$, where y =$\left(\begin{array}{cc}a& 0\\ 0& 0\end{array}\right)$; - (3)
- if x∈ S
_{2}and y∈S_{1}, x + y = y + x (see (2)); - (4)
- if x = y∈S
_{2}, x + y =$\left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right)$; - (5)
- if x,y∈S
_{2}and x ≠ y, x + y =$\left(\begin{array}{cc}2& 0\\ 0& 0\end{array}\right)$.

**Example**

**15.**

**Theorem**

**8.**

- (1)
- If e is aquasi-rightunit element of S and ee = a, a$\in $S, then ex = ax for all x$\in $S.
- (2)
- The quasi-right unite elementis unique in S.

**Proof.**

**Theorem**

**9.**

- (1)
- Let S be a variant CA-groupoid and e is the quasi-right unite element on S, then S
_{1}= S−{e} is a commutative semigroup. - (2)
- Let S be a commutative monoid with unit element e and a is an element such that {a}${{\displaystyle \cap}}^{\text{}}$S = $\varnothing $, then S
_{2}= S${{\displaystyle \cup}}^{\text{}}${a} is a variant CA-groupoid if define xa = x, ax = ex, aa = e, for all x$\in $S.

**Proof.**

_{1}= S−{e} such that xy = e, then for all a∈S−{e}, a*xy = ae = a, so we have:

_{1}= S−{e}, applying Theorem 8 (1), ex = (ee)x, and:

_{2}= S${{\displaystyle \cup}}^{\text{}}${a}. Define a new binary operation • on S

_{2}:

_{2}, if x, y∈ S, then x• y = x*y; if x∈ S, then x•a = x, a•x = e•x, a•a = e.

_{2}, •) is a groupoid. For all x, y, z$\in $S, by the definition of operation • we have:

_{2}, •) is a variant CA-groupoid with the quasi-right unit element a. □

**Proposition**

**7.**

**Theorem**

**10.**

- (1)
- If b = a, then {e, a} is a variant sub CA-groupoid of S;
- (2)
- If b$\ne $a, then {e, a, b} is a variant sub CA-groupoid of S.

**Proof.**

**Theorem**

**11.**

_{1}, *

_{1}) and (S

_{2}, *

_{2}) be two variant CA-groupoids, e

_{1}and e

_{2}are quasi-right unit elements of (S

_{1}, *

_{1}) and (S

_{2}, *

_{2}), S

_{1}${{\displaystyle \cap}}^{\text{}}$S

_{2}= {e} (e = e

_{1}= e

_{2}). Denote S = S

_{1}${{\displaystyle \cup}}^{\text{}}$S

_{2}, and define the operation * on S as follows:

_{1}, then a*b = a*

_{1}b;

_{2}, then a*b = a*

_{2}b;

_{1}-{e}, b$\in $S

_{2}-{e}, then a*b = b;

_{2}-{e}, b$\in $S

_{1}-{e}, then a*b = a.

**Proof.**

- (1)
- If a, b, c$\in $S
_{1}, or a, b, c$\in $S_{2}, then a*(b*c) = c*(a*b); - (2)
- If a$\in $S
_{1}-{e}, b$\in $S_{2}-{e} and c$\in $S_{2}-{e}, then a*(b*c) = b*c = c*b = c*(a*b); - (3)
- If a$\in $S
_{2}-{e}, b$\in $S_{1}-{e} and c$\in $S_{2}-{e}, then a*(b*c) = a*c = c*a = c*(a*b); - (4)
- If a$\in $S
_{2}-{e}, b$\in $S_{2}-{e} and c$\in $S_{1}-{e}, then a*(b*c) = a*b = c*(a*b); - (5)
- If a$\in $S
_{1}-{e}, b$\in $S_{1}-{e} and c$\in $S_{2}-{e}, then a*(b*c) = a*c = c = c*(a*b); - (6)
- If a$\in $S
_{1}-{e}, b$\in $S_{2}-{e} and c$\in $S_{1}-{e}, then a*(b*c) = a*b = c*(a*b); - (7)
- If a$\in $S
_{2}-{e}, b$\in $S_{1}-{e} and c$\in $S_{1}-{e}, then a*(b*c) = a = a*b = c*(a*b).

**Example**

**16.**

**Proposition**

**8.**

_{1}, *

_{1}) and (S

_{2}, *

_{2}) be two variant CA-groupoids, e

_{1}and e

_{2}are variant unit elements of (S

_{1}, *

_{1}) and (S

_{2}, *

_{2}), S

_{1}${{\displaystyle \cap}}^{\text{}}$S

_{2}=$\varnothing $and S

_{2}is commutative. Denote S = S

_{1}${{\displaystyle \cup}}^{\text{}}$S

_{2}, and define the operation * in S as follows:

- (1)
- if a, b$\in $S
_{1}, then a*b = a*_{1}b; - (2)
- if a, b$\in $S
_{2}, then a*b = a*_{2}b; - (3)
- if a$\in $S
_{1}, b$\in $S_{2}, then a*b = b; - (4)
- if a$\in $S
_{2}, b$\in $S_{1}, then a*b = a.

_{1}.

**Example**

**17.**

**Theorem**

**12.**

_{1}be a variant CA-groupoid with order n (n ≥ 2 and n is an even number) and the quasi- right unit element e

_{1}$\in $S

_{1}, let S

_{2}be a variant CA-groupoid with order 2 and the quasi-right unit element e

_{2}$\in $S

_{2}. If S = S

_{1}${{\displaystyle \cup}}^{\text{}}$S

_{2}and S

_{1}${{\displaystyle \cap}}^{\text{}}$S

_{2}=$\varnothing $, then S is a variant CA-groupoid, when it such that any of the following conditions:

- (1)
- for thevariantCA-groupoid S, thequasi-rightunit element e = e
_{1}, and e_{2}*e_{1}= e_{2}, and for all x$\in $S, x*(e_{2}* e_{2}) = (e_{2}* e_{2})*x = e_{2}* e_{2}, x* e_{2}= e_{2}* e_{2}, e_{2}*x = e_{2}* e_{2}(x$\ne $e_{1}); - (2)
- for thevariantCA-groupoid S, thequasi-rightunit element e = e
_{1}and for all x$\in $S, x*(e_{2}* e_{2}) = (e_{2}* e_{2})*x = e_{2}* e_{2}, x* e_{2}= e_{2}*x = e_{2}.

**Proof.**

_{1}, x*yz = z*xy = y*zx, and:

_{2}= x*e

_{2}e

_{2}= e

_{2}e

_{2}, e

_{2}*xy = e

_{2}e

_{2}(xy ≠ e

_{1})

_{2}= e

_{2}*xy = y*e

_{2}x. Denote e

_{2}e

_{2}= b, then:

_{2}e

_{2 }= xb = b, e

_{2}*xe

_{2 }= e

_{2}*e

_{2}e

_{2 }= b,

_{2}e

_{2}= e

_{2}*xe

_{2}= e

_{2}*e

_{2}x. And:

_{2}= x*e

_{2}b = xb = b, b*xe

_{2}= b*e

_{2}e

_{2}= b = e

_{2}b = e

_{2}*bx, e

_{2}*xb = e

_{2}b = b,

_{2}= e

_{2}*xb = b*e

_{2}x, and x*e

_{2}b = b*xe

_{2}= e

_{2}*bx. Obviously, x*bb = b*xb = b*bx. Hence, S is a variant CA-groupoid.

_{1}, x*yz = z*xy = y*zx, and:

_{2}= xe

_{2}= e

_{2}, e

_{2}*xy = e

_{2}, y*e

_{2}x = ye

_{2}= e

_{2}.

_{2}= e

_{2}*xy = y*e

_{2}x. Assume e

_{2}e

_{2}= b, then:

_{2}e

_{2}= xb = b, e

_{2}*xe

_{2}= e

_{2}*e

_{2}x = e

_{2}e

_{2}= b.

_{2}e

_{2}= e

_{2}*xe

_{2}= e

_{2}*e

_{2}x. Moreover:

_{2}= x*e

_{2}b = xb = b, b*xe

_{2}= be

_{2}= b = e

_{2}b = e

_{2}*bx, e

_{2}*xb = e

_{2}b = b = be

_{2}= b*e

_{2}x.

_{2}= e

_{2}*xb = b*e

_{2}x, and x*e

_{2}b = b*xe

_{2}= e

_{2}*bx. Obviously, x*bb = b*xb = b*bx. Hence, S is a variant CA-groupoid. □

## 6. Conclusions

- (1)
- Every left cancellative element in CA-groupoid is right cancellative (see Theorem 1);
- (2)
- For a CA-groupoid, it is left cancellative if and only if it is right cancellative (see Theorem 1 and Corollary 1);
- (3)
- For a CA-groupoid, it is left separative if and only if it is right separative, and if and only if it is quasi-separative (see Theorem 4 and Corollary 1);
- (4)
- Every left quasi-cancellative CA-groupoid is right quasi-cancellative (see Theorem 5); every power cancellative CA-groupoid is separative (see Theorem 7);
- (5)
- For a variant CA-groupoid, its quasi-right unit element is unique;
- (6)
- A variant CA-groupoid can be decomposed into the quasi-right unit element and a commutative CA-groupoid; starting from any commutative semigroup, one can construct a variant CA-groupoid (see Theorem 9);
- (7)
- There are many ways to construct a new variant CA-groupoid from the existing variant CA-groupoids (see Theorems 11 and 12).

## Author Contributions

## Funding

## Conflicts of Interest

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* | $\overline{0}$ | $\overline{1}$ | $\overline{2}$ | $\overline{3}$ | $\overline{4}$ |
---|---|---|---|---|---|

$\overline{\mathbf{0}}$ | $\overline{0}$ | $\overline{1}$ | $\overline{2}$ | $\overline{3}$ | $\overline{4}$ |

$\overline{\mathbf{1}}$ | $\overline{1}$ | $\overline{2}$ | $\overline{3}$ | $\overline{4}$ | $\overline{0}$ |

$\overline{\mathbf{2}}$ | $\overline{2}$ | $\overline{3}$ | $\overline{4}$ | $\overline{0}$ | $\overline{1}$ |

$\overline{\mathbf{3}}$ | $\overline{3}$ | $\overline{4}$ | $\overline{0}$ | $\overline{1}$ | $\overline{2}$ |

$\overline{\mathbf{4}}$ | $\overline{4}$ | $\overline{0}$ | $\overline{1}$ | $\overline{2}$ | $\overline{3}$ |

* | 1 | 2 | 3 | 4 |
---|---|---|---|---|

1 | 4 | 3 | 2 | 1 |

2 | 3 | 1 | 4 | 2 |

3 | 2 | 4 | 1 | 3 |

4 | 1 | 2 | 3 | 4 |

* | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

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

2 | 1 | 1 | 2 | 1 | 2 |

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

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

5 | 1 | 1 | 1 | 1 | 1 |

* | 1 | 2 | 3 | 4 |
---|---|---|---|---|

1 | 1 | 2 | 4 | 3 |

2 | 2 | 1 | 3 | 4 |

3 | 3 | 4 | 4 | 3 |

4 | 4 | 3 | 3 | 4 |

* | 1 | 2 | 3 | 4 |
---|---|---|---|---|

1 | 4 | 2 | 1 | 1 |

2 | 2 | 2 | 2 | 2 |

3 | 1 | 2 | 3 | 4 |

4 | 1 | 2 | 4 | 4 |

* | 1 | 2 | 3 | 4 |
---|---|---|---|---|

1 | 4 | 3 | 2 | 1 |

2 | 3 | 2 | 3 | 2 |

3 | 2 | 3 | 2 | 3 |

4 | 1 | 2 | 3 | 4 |

* | 1 | 2 | 3 | 4 |
---|---|---|---|---|

1 | 1 | 2 | 3 | 1 |

2 | 1 | 4 | 3 | 2 |

3 | 3 | 3 | 3 | 3 |

4 | 1 | 2 | 3 | 4 |

* | 1 | 2 | 3 | 4 |
---|---|---|---|---|

1 | 1 | 1 | 1 | 1 |

2 | 1 | 2 | 1 | 1 |

3 | 1 | 1 | 3 | 4 |

4 | 1 | 1 | 4 | 3 |

* | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

1 | 3 | 3 | 5 | 3 | 3 |

2 | 3 | 3 | 5 | 3 | 3 |

3 | 4 | 4 | 3 | 5 | 5 |

4 | 3 | 3 | 5 | 3 | 3 |

5 | 3 | 3 | 5 | 3 | 3 |

* | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

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

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

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

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

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

* | 1 | 2 | 3 | 4 |
---|---|---|---|---|

1 | 1 | 2 | 3 | 4 |

2 | 2 | 1 | 4 | 3 |

3 | 3 | 4 | 2 | 1 |

4 | 4 | 3 | 1 | 2 |

* | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

1 | 3 | 3 | 3 | 4 | 4 |

2 | 2 | 3 | 3 | 4 | 4 |

3 | 3 | 3 | 3 | 4 | 4 |

4 | 4 | 4 | 4 | 4 | 4 |

5 | 5 | 4 | 4 | 4 | 4 |

+ | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

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

2 | 2 | 3 | 3 | 4 | 4 |

3 | 3 | 3 | 3 | 4 | 4 |

4 | 4 | 4 | 4 | 4 | 4 |

5 | 5 | 4 | 4 | 4 | 4 |

* | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

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

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

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

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

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

+ | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

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

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

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

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

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

* | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

1 | 3 | 3 | 3 | 4 | 4 | 6 |

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

3 | 3 | 3 | 3 | 4 | 4 | 6 |

4 | 4 | 4 | 4 | 4 | 4 | 6 |

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

6 | 6 | 6 | 6 | 6 | 6 | 6 |

+ | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

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

2 | 1 | 2 | 3 | 3 | 3 | 2 |

3 | 1 | 3 | 3 | 3 | 3 | 3 |

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

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

6 | 1 | 2 | 3 | 4 | 5 | 6 |

+ | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

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

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

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

4 | 4 | 4 | 4 | 4 | 5 | 4 |

5 | 5 | 5 | 5 | 5 | 4 | 5 |

6 | 1 | 2 | 3 | 4 | 5 | 6 |

*_{1} | 1 | 2 | 3 | 4 |
---|---|---|---|---|

1 | 2 | 2 | 4 | 4 |

2 | 2 | 2 | 4 | 4 |

3 | 3 | 4 | 4 | 4 |

4 | 4 | 4 | 4 | 4 |

*_{2} | 1 | 5 | 6 | 7 |
---|---|---|---|---|

1 | 5 | 5 | 7 | 7 |

5 | 5 | 5 | 7 | 7 |

6 | 6 | 7 | 5 | 5 |

7 | 7 | 7 | 5 | 5 |

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

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

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

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

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

5 | 5 | 5 | 5 | 5 | 5 | 7 | 7 |

6 | 6 | 6 | 6 | 6 | 7 | 5 | 5 |

7 | 7 | 7 | 7 | 7 | 7 | 5 | 5 |

*_{1} | 1 | 2 | 3 | 4 |
---|---|---|---|---|

1 | 3 | 2 | 2 | 4 |

2 | 2 | 2 | 2 | 4 |

3 | 3 | 2 | 2 | 4 |

4 | 4 | 4 | 4 | 4 |

*_{2} | 5 | 6 | 7 | 8 |
---|---|---|---|---|

5 | 8 | 6 | 7 | 8 |

6 | 6 | 6 | 6 | 6 |

7 | 7 | 6 | 6 | 7 |

8 | 8 | 6 | 7 | 8 |

* | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|

1 | 3 | 2 | 2 | 4 | 5 | 6 | 7 | 8 |

2 | 2 | 2 | 2 | 4 | 5 | 6 | 7 | 8 |

3 | 3 | 2 | 2 | 4 | 5 | 6 | 7 | 8 |

4 | 4 | 4 | 4 | 2 | 5 | 6 | 7 | 8 |

5 | 5 | 5 | 5 | 5 | 8 | 6 | 7 | 8 |

6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 |

7 | 7 | 7 | 7 | 7 | 7 | 6 | 6 | 7 |

8 | 8 | 8 | 8 | 8 | 8 | 6 | 7 | 8 |

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## Share and Cite

**MDPI and ACS Style**

Ma, Z.; Zhang, X.; Smarandache, F.
Some Results on Various Cancellative CA-Groupoids and Variant CA-Groupoids. *Symmetry* **2020**, *12*, 315.
https://doi.org/10.3390/sym12020315

**AMA Style**

Ma Z, Zhang X, Smarandache F.
Some Results on Various Cancellative CA-Groupoids and Variant CA-Groupoids. *Symmetry*. 2020; 12(2):315.
https://doi.org/10.3390/sym12020315

**Chicago/Turabian Style**

Ma, Zhirou, Xiaohong Zhang, and Florentin Smarandache.
2020. "Some Results on Various Cancellative CA-Groupoids and Variant CA-Groupoids" *Symmetry* 12, no. 2: 315.
https://doi.org/10.3390/sym12020315