# Involution Abel–Grassmann’s Groups and Filter Theory of Abel–Grassmann’s Groups

^{*}

## Abstract

**:**

## 1. Introduction

**R**

^{3}with multiplication given by the vector cross product is an example of an algebra that is not associative, at the same time; Jordan algebra and Lie algebra are non-associative.

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Proposition**

**2.**

**Proposition**

**3.**

**Proposition**

**4.**

**Proposition**

**5.**

- (1)
- N is an AG-group.
- (2)
- Every element of N has a right inverse.
- (3)
- Every element a of N has a unique inverse a
^{−1}. - (4)
- The equation x*a = b has a unique solution for all a, b$\in $N.

**Proposition**

**6.**

## 3. Some Examples and New Results of AG-Groups

**Example**

**1.**

_{a}, φ

_{b}and φ

_{c}, respectively (see Figure 1). There is of course the movement that does nothing, which is denoted by φ

_{e}. The following figure gives an intuitive description of these transformations. Denote N = {φ

_{e}, φ

_{a}, φ

_{b}, φ

_{c}}.

_{e}φ

_{e}= φ

_{e}, φ

_{a}φ

_{c}= φ

_{c}φ

_{a}= φ

_{e}, φ

_{b}φ

_{b}= φ

_{e}. That is, φ

_{e}

^{−1}= φ

_{e}, φ

_{a}

^{−1}= φ

_{c}, φ

_{b}

^{−1}= φ

_{b},φ

_{c}

^{−1}= φ

_{a}. Now, we define operations * on N as follows:

_{x}*φ

_{y}= φ

_{x}

^{−}

^{1}φ

_{y}, ∀x, y∈{e, a, b, c}.

**Example**

**2.**

**R**−{0}}, where

**R**represents the set of all real numbers. Define binary operation * as follows:

**Example**

**3.**

**R**, b = 1 or −1}, where

**R**represents the set of all real numbers. Define binary operation * as follows:

**Example**

**4.**

**R**, b = 1, −1, i, or −i}, where

**R**represents the set of all real numbers and I represents the imaginary unit. Define binary operation * as follows:

**Definition**

**3.**

- (1)
- e*x
^{−1}= x^{−1}*e, for all x in N; - (2)
- e*x = x*e, for all x in N; or
- (3)
- x
^{−1}*y^{−1}= y^{−1}*x^{−1}, for all x, y in N.

**Theorem**

**1.**

**Proof.**

^{−1}= x

^{−1}*e.

**Theorem**

**2.**

- (1)
- x = x * y and${y}^{2}$= y*x imply that x = y; and
- (2)
- x = y * x and${y}^{2}$= x * y imply that x = y.

**Proof.**

**Definition**

**4.**

- (1)
- a*(a*b) = (a*a)*b, ∀a, b$\in $N; or
- (2)
- a*(b*b) = (a*b)*b, ∀a, b$\in $N.

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

- (1)
- U(N) is sub-algebra of N.
- (2)
- U(N) is maximal subgroup of N with identity e.

**Proof.**

**Theorem**

**5.**

- (1)
- P(N) is the subalgebra of N;
- (2)
- f is a homomorphism mapping from N to P(N), where f: N→P(N), f(x)=x*x∈P(N).

**Proof.**

## 4. Involution AG-Groups and Generalized Involution AG-Groups

**Definition**

**5.**

**Example**

**5.**

**Example**

**6.**

**Theorem**

**6.**

- (1)
- P(N) = {e}, where P(N) is defined as Theorem 5.
- (2)
- (x*x)*x = x for any x∈N.

**Proof.**

**Theorem**

**7.**

**Proof.**

^{−1}= x

^{−1}∘x = e.

**Definition**

**6.**

**Example**

**7.**

**Theorem**

**8.**

- (1)
- $\approx $is an equvalent relation on N, and we denote the equivalent class contained x by${\left[x\right]}_{\approx}$.
- (2)
- The equivalent class contained e by${\left[e\right]}_{\approx}$is an involution sub-AG-group.
- (3)
- For any x, y, z$\in $N, x$\approx $y implies x*z$\approx $y*z and z*x$\approx $z*y.
- (4)
- The quotient (N/$\approx $, *) is an involution AG-group.

**Proof.**

**Theorem**

**9.**

**Proof.**

## 5. Filter of AG-Groups and Homomorphism Theorems

**Definition**

**7.**

- (1)
- e$\in $F;
- (2)
- x*x$\in $F; and
- (3)
- x$\in $F and x*y$\in $F imply that y$\in $F.

**Theorem**

**10.**

**Proof.**

**Theorem**

**11.**

- (1)
- ${\approx}_{F}$is an equivalent relation on N.
- (2)
- x${\approx}_{F}$y and a${\approx}_{F}$b imply x*a${\approx}_{F}$y*b.
- (3)
- f: N$\to $N/F is a homomorphism mapping, where N/F = {${\left[x\right]}_{F}$: x$\in $N},${\left[x\right]}_{F}$denote the equivalent class contained x.

**Proof.**

**Theorem**

**12.**

**Proof.**

**Theorem**

**13.**

**Proof.**

^{−1}*b, where a

^{−1}is the left inverse of a in N. Then,

^{−1}*b)*(a

^{−1}*b) = (a

^{−1}*a

^{−1})*(b*b) = (a

^{−1}*a

^{−1})*(x*y) = (a

^{−1}*a

^{−1})*((a*a)*y) = (a

^{−1}*a

^{−1})*((y*a)*a) = (a

^{−1}*(y*a))*(a

^{−1}*a) = (a

^{−1}*(y*a))*e = (e*(y*a))*a

^{−1}= (y*a)*a

^{−1}= (a

^{−1}*a)*y = e*y= y.

**Theorem**

**14.**

**Proof.**

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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* | φ_{e} | φ_{a} | φ_{b} | φ_{c} |
---|---|---|---|---|

φ_{e} | φ_{e} | φ_{a} | φ_{b} | φ_{c} |

φ_{a} | φ_{c} | φ_{e} | φ_{a} | φ_{b} |

φ_{b} | φ_{b} | φ_{c} | φ_{e} | φ_{a} |

φ_{c} | φ_{a} | φ_{b} | φ_{c} | φ_{e} |

* | a | b | c | d |
---|---|---|---|---|

a | a | b | c | d |

b | b | a | d | c |

c | d | c | a | b |

d | c | d | b | a |

* | e | a | b | c |
---|---|---|---|---|

e | e | a | b | c |

a | a | e | c | b |

b | c | b | a | e |

c | b | c | e | a |

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**MDPI and ACS Style**

Zhang, X.; Wu, X.
Involution Abel–Grassmann’s Groups and Filter Theory of Abel–Grassmann’s Groups. *Symmetry* **2019**, *11*, 553.
https://doi.org/10.3390/sym11040553

**AMA Style**

Zhang X, Wu X.
Involution Abel–Grassmann’s Groups and Filter Theory of Abel–Grassmann’s Groups. *Symmetry*. 2019; 11(4):553.
https://doi.org/10.3390/sym11040553

**Chicago/Turabian Style**

Zhang, Xiaohong, and Xiaoying Wu.
2019. "Involution Abel–Grassmann’s Groups and Filter Theory of Abel–Grassmann’s Groups" *Symmetry* 11, no. 4: 553.
https://doi.org/10.3390/sym11040553