# The Structure Theorems of Pseudo-BCI Algebras in Which Every Element is Quasi-Maximal

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1**

**.**An algebra (A; →, 1) of type (2,0) is called a BCI-algebra if the following conditions are satisfied for all x, y, z from A:

- (1)
- $x\to y\le \left(y\to z\right)\to \left(x\to z\right),$
- (2)
- $x\le \left(x\to y\right)\to y,$
- (3)
- $x\le x,$
- (4)
- $x\le y,y\le ximplyx=y,$where$x\le y$means $x\to y=1$. An algebra (A; →, 1) of type (2,0) is called a BCK-algebra if it is a BCI-algebra and satisfies:
- (5)
- $x\to 1=1,$∀x ∈ A.

**Definition**

**2**

**Definition**

**3**

**Definition**

**4**

**.**A pseudo-BCK algebra is a structure (A; $\le $, $\to $, $\u21dd$, 1), where “$\le $” is a binary relation on A, “$\to $” and “$\u21dd$” are binary operations on A and “1” is an element of A, verifying the axioms: for all x, y, z $\in $ A,

- (1)
- $x\to y\le $($y\to z$)$\u21dd$($x\to z$),$x\u21ddy\le $($y\u21ddz$)$\to $($x\u21ddz$),
- (2)
- $x\le \left(x\to y\right)\u21ddy,x\le \left(x\u21ddy\right)\to y$
- (3)
- $x\le x,$
- (4)
- $x\le 1,$
- (5)
- $x\le y,$$y\le x\Rightarrow $x = y,
- (6)
- $x\le y\u21ddx\to y=1\iff x\u21ddy=1.$

**Proposition**

**1**

**.**Let (A; $\le $, $\to $, $\u21dd$, 1) be a pseudo-BCK algebra, then A satisfy the following properties ($\forall x,y,z\in A$):

- (1)
- $x\le y\Rightarrow y\to z\le $$x\to z,$$y\u21ddz\le $$x\u21ddz$
- (2)
- $x\le y,$$y\le z$$\Rightarrow $x$\le $z,
- (3)
- $x\u21dd\left(y\to z\right)=y\to \left(x\u21ddz\right),$
- (4)
- $x\le y\to z\iff y\le x\u21ddz,$
- (5)
- $x\to y\le \left(z\to x\right)\to \left(z\to y\right),$$x\u21ddy\le \left(z\u21ddx\right)\u21dd\left(z\u21ddy\right),$
- (6)
- $x\le y\to x,x\le y\u21ddx,$
- (7)
- $1\to x=x,\text{}1\u21ddx=x,$
- (8)
- $x\le y\Rightarrow z\to x\le $$z\to y,$$z\u21ddx\le $$z\u21ddy,$
- (9)
- $\left(\left(y\to x\right)\u21ddx\right)\to x=y\to x,$$\left(\left(y\u21ddx\right)\to x\right)\u21ddx=y\u21ddx.$

**Definition**

**5**

**.**A pseudo-BCI algebra is a structure (A; $\le $, $\to $, $\u21dd$, 1), where “$\le $” is a binary relation on A, “$\to $” and “$\u21dd$” are binary operations on A and “1” is an element of A, verifying the axioms: for all x, y, z $\in $ A,

- (1)
- $x\to y\le \left(y\to z\right)\u21dd\left(x\to z\right),$$x\u21ddy\le \left(y\u21ddz\right)\to \left(x\u21ddz\right),$
- (2)
- $x\le \left(x\to y\right)\u21ddy,x\le \left(x\u21ddy\right)\to y,$
- (3)
- $x\le x,$
- (4)
- if$x\le y$and $y\le x$, then x = y,
- (5)
- $x\le y$iff$x\to y=1$iff$x\u21ddy=1.$

**Proposition**

**2**

**.**Let (A; $\le $, $\to $, $\u21dd$, 1) be a pseudo-BCI algebra, then A satisfy the following properties ($\forall x,y,z\in A$):

- (1)
- if$1\le x,$then$x=1,$
- (2)
- if$x\le y,$then$y\to z\le $$x\to z$and$y\u21ddz\le $$x\u21ddz,$
- (3)
- if$x\le y$and$y\le z,$then $x\le z$,
- (4)
- $x\u21dd\left(y\to z\right)=y\to \left(x\u21ddz\right),$
- (5)
- $x\le y\to z,$iff$y\le x\u21ddz$
- (6)
- $x\to y\le \left(z\to x\right)\to \left(z\to y\right),$$x\u21ddy\le \left(z\u21ddx\right)\u21dd\left(z\u21ddy\right),$
- (7)
- if$x\le y,$then$z\to x\le $$z\to y$and$z\u21ddx\le $$z\u21ddy,$
- (8)
- $1\to x=x,1\u21ddx=x,$
- (9)
- $\left(\left(y\to x\right)\u21ddx\right)\to x=y\to x,$$\left(\left(y\u21ddx\right)\to x\right)\u21ddx=y\u21ddx,$
- (10)
- $x\to y\le \left(y\to x\right)\u21dd1,$$x\u21ddy\le \left(y\u21ddx\right)\to 1,$
- (11)
- (x → y) → 1 = (x → 1) ⇝ (y → 1),(x ⇝ y) ⇝ 1 = (x ⇝ 1) → (y → 1)
- (12)
- $x\to 1=x\u21dd1$.

**Definition**

**6**

**Proposition**

**3**

- (G1) for all x, y, z ∈ A, (x $\to $ y) $\to $ (x $\to $ z) = y $\to $ z and
- (G2) for all x, y, z ∈ A, (x $\u21dd$ y) $\u21dd$ (x $\u21dd$ z) = y $\u21dd$ z.

**Proposition**

**4**

**.**Let

**A**= (A; $\le $, $\to $, $\u21dd$, 1) be an anti-grouped pseudo-BCI algebra. Define

**Φ(A)**= (A; +, −, 1) by

**Φ(A)**is a group. Conversely, let

**G**= (G; +, −, 1) be a group. Define

**Ψ(G)**= (G;$\le $,$\to $, $\u21dd$, 1), where

**Ψ(G)**is an anti-grouped pseudo-BCI algebra. Moreover, the mapping

**Φ**and

**Ψ**are mutually inverse.

**Definition**

**7**

**.**Let (A; $\le $, $\to $, $\u21dd$, 1) be a pseudo-BCI algebra. Denote

**Definition**

**8**

**Proposition**

**5**

## 3. Some New Concepts and Results

**Lemma**

**1**

**.**Let (X; ${\to}_{X}$, $\u21dd{}_{X},\text{}{1}_{X}$) and (Y; ${\to}_{Y}$, $\u21dd{}_{Y},\text{}{1}_{Y}$) be two pseudo-BCI algebras. Define two binary operators $\to $, $\u21dd$ on $X\times Y$ as follwos: for any (x

_{1}, y

_{1}), (x

_{2}, y

_{2}) ∈ $X\times Y,$

_{X}, 1

_{Y}). Then ($X\times Y$; $\to $, $\u21dd$, 1) is a pseudo-BCI algebra.

**Lemma**

**2.**

**Definition**

**9.**

**Definition**

**10.**

**Definition**

**11.**

**Theorem**

**1.**

- (a1)
- $\forall x\in A$, x is a quasi-maximal element;
- (a2)
- $\forall x\in A$,$y\in A-\left\{1\right\},$$x\le y$implies $x=y$;
- (a3)
- $\forall x\in A,$x is a quasi-left unit elemen w.r.t$\to ,\u21dd$, that is,$x\ne y$implies$x\to y=y$and $x\u21ddy=y$;
- (a4)
- $\forall x,y\in A$,$x\ne y$implies $x\to y=y$;
- (a5)
- $\forall x,y\in A,x\ne y$implies $x\u21ddy=y$.

**Proof.**

- (i)
- when $\left(x\to y\right)\u21ddy=1$, we can get $x\to y\le y\le x\to y$, that is, $y=x\to y$;
- (ii)
- when $\left(x\to y\right)\u21ddy\ne 1$, from this and $x\le \left(x\to y\right)\u21ddy,$ using (a2) we have $x=\left(x\to y\right)\u21ddy.$ Combine the aforementioned conclusion $x=y\to x$, we can get$$x=y\to x=y\to \left(\left(x\to y\right)\u21ddy\right)=\left(x\to y\right)\u21dd\left(y\to y\right)=\left(x\to y\right)\u21dd1=1,$$

**Corollary**

**1.**

## 4. The Class of Pseudo-BCI Algebras in Which Every Element is Quasi-Maximal

**Example**

**1.**

**Definition**

**12.**

**Theorem**

**2.**

**Proof.**

**Lemma**

**3**

**.**Let A be a pseudo-BCI algebra, K(A) the pseudo-BCK part of A. If AG(A) = (A − K(A))$\cup ${1} is subalgebra of A, then ($\forall x,y\in A$)

- (1)
- If$x\in K\left(A\right)$and$y\in A-K\left(A\right),$then$x\to y=x\u21ddy=y.$
- (2)
- If$x\in A-K\left(A\right)$and$y\in K\left(A\right),$then$x\to y=x\u21ddy=x\to 1.$

**Lemma**

**4.**

- (1)
- for any x, y in A, $x\le $ y implies x = y;
- (2)
- for any x, y in A,$x=\left(x\to y\right)\u21ddy=\left(x\u21ddy\right)\to y.$

**Theorem**

**3.**

- (1)
- A is a QM-pseudo-BCI algebra;
- (2)
- K(A) is quasi-alternating BCK-algebras and AG(A) = (A − K(A))$\cup ${1};
- (3)
- $\forall x,y\in A,$$x\ne y$implies$\left(x\to y\right)\u21ddy=\left(x\to 1\right)\to 1;$
- (4)
- $\forall x,y\in A,$$x\ne y$implies$\left(x\u21ddy\right)\to y=\left(x\to 1\right)\to 1;$
- (5)
- $\forall x,y\in A,$$x\ne y$implies$\left(x\to y\right)\to y=\left(x\to 1\right)\to 1;$
- (6)
- $\forall x,y\in A,$$x\ne y$implies$\left(x\u21ddy\right)\u21ddy=\left(x\to 1\right)\to 1.$

**Proof.**

**Theorem**

**4.**

**Proof.**

## 5. Weak Associative Pseudo-BCI Algebras

**Definition**

**13.**

**Example**

**2**

**Theorem**

**5.**

**Proof.**

**Example**

**3.**

**Lemma**

**5**

- (1)
- A is associative, that is,$\left(x\to y\right)\to z=x\to \left(y\to z\right)$for any x, y, z in A;
- (2)
- for any x in A,$x\to 1=x;$
- (3)
- for all x, y in A,x$\to y=y\to x.$

**Theorem**

**6.**

- (1)
- K(A) is quasi-alternating BCK-algebra andAG(A) = (A − K(A))$\cup ${1};
- (2)
- For any x in AG(A), x$\to 1=x\u21dd1=x$;
- (3)
- For any x, y in A, x$\to y=x\u21ddy,$ that is, A is a BCI-algebra;
- (4)
- AG(A) is an Abel group, that is, AG(A) is associative BCI-algebra.

**Proof.**

- (i)
- when x, y in K(A), by (1), K(A) is a BCK-algebra, so x $\to y=x\u21ddy;$
- (ii)
- when x, y in (A − K(A)), by (1) and (2), applying Proposition 2 (11),$$x\to y=\left(x\to y\right)\to 1=\left(x\to 1\right)\u21dd\left(y\to 1\right)=x\u21ddy;$$
- (iii)
- when x in K(A), y in (A − K(A)), using Lemma 3 (1), x $\to y=x\u21ddy;$
- (iv)
- when y in K(A), x in (A − K(A)), using Lemma 3 (2), x $\to y=x\u21ddy;$

**Theorem**

**7.**

**Theorem**

**8.**

- (1)
- for any$x,y,z\in A,$$\left(x\to y\right)\to z=x\to \left(y\to z\right)$when$\left(x\ne y,x\ne z\right);$
- (2)
- for any$x,y,z\in A,$$\left(x\u21ddy\right)\u21ddz=x\u21dd\left(y\u21ddz\right)$when$\left(x\ne y,x\ne z\right);$
- (3)
- for any$x,y,z\in A,$$\left(x\to y\right)\u21ddz=x\to \left(y\u21ddz\right)$when$\left(x\ne y,x\ne z\right);$
- (4)
- for any$x,y,z\in A,$$\left(x\u21ddy\right)\to z=x\u21dd\left(y\to z\right)$when$\left(x\ne y,x\ne z\right).$

**Proof.**

_{qm}(A)∪AG(A) is a WA-pseudo-BCI subalgebra of A, where K

_{qm}(A) is the set of all quasi-maximal element in K(A). For example, {c, d, 1} is a WA-pseudo-BCI subalgebra of the pseudo-BCI algebra A in Example 3.

_{qm}(A)∪AG(A) is a QM-pseudo-BCI subalgebra of A, where K

_{qm}(A) is the set of all quasi-maximal element in K(A).

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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→ | a | b | c | d | e | f | g | 1 |
---|---|---|---|---|---|---|---|---|

a | 1 | b | c | d | e | f | g | 1 |

b | a | 1 | c | d | e | f | g | 1 |

c | c | c | 1 | f | g | d | e | c |

d | d | d | e | 1 | c | g | f | d |

e | f | f | g | c | 1 | e | d | f |

f | e | e | d | g | f | 1 | c | e |

g | g | g | c | d | e | f | 1 | g |

1 | a | b | c | d | e | f | g | 1 |

⇝ | a | b | c | d | e | f | g | 1 |
---|---|---|---|---|---|---|---|---|

a | 1 | b | c | d | e | f | g | 1 |

b | a | 1 | c | d | e | f | g | 1 |

c | c | c | 1 | f | g | d | e | c |

d | d | d | e | 1 | c | g | f | d |

e | f | f | g | c | 1 | e | d | f |

f | e | e | d | g | f | 1 | c | e |

g | g | g | c | d | e | f | 1 | g |

1 | a | b | c | d | e | f | g | 1 |

→ | a | b | c | d | e | f | 1 |
---|---|---|---|---|---|---|---|

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

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

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

d | d | d | d | 1 | f | e | d |

e | e | e | e | f | 1 | d | e |

f | f | f | f | e | d | 1 | f |

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

→ | a | b | c | d | 1 |
---|---|---|---|---|---|

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

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

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

d | d | d | d | 1 | d |

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

⇝ | a | b | c | d | 1 |
---|---|---|---|---|---|

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

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

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

d | d | d | d | 1 | d |

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

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**MDPI and ACS Style**

Wu, X.; Zhang, X.
The Structure Theorems of Pseudo-BCI Algebras in Which Every Element is Quasi-Maximal. *Symmetry* **2018**, *10*, 465.
https://doi.org/10.3390/sym10100465

**AMA Style**

Wu X, Zhang X.
The Structure Theorems of Pseudo-BCI Algebras in Which Every Element is Quasi-Maximal. *Symmetry*. 2018; 10(10):465.
https://doi.org/10.3390/sym10100465

**Chicago/Turabian Style**

Wu, Xiaoying, and Xiaohong Zhang.
2018. "The Structure Theorems of Pseudo-BCI Algebras in Which Every Element is Quasi-Maximal" *Symmetry* 10, no. 10: 465.
https://doi.org/10.3390/sym10100465