# Some Implicativities for Groupoids and BCK-Algebras

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

- (I)
- $x\ast x=0$,
- (II)
- $0\ast x=0$,
- (III)
- $x\ast y=0$ and $y\ast x=0$ imply $x=y$ for all $x,y\in X$.

- (IV)
- $\left(\right(x\ast y)\ast (x\ast z\left)\right)\ast (z\ast y)=0$,
- (V)
- $(x\ast (x\ast y\left)\right)\ast y=0$ for all $x,y,z\in X$.

**Theorem**

**1**

**.**If $(X,\ast ,0)$ is a $BCK$-algebra, then

**Example**

**1.**

**Theorem**

**2**

**.**$\left(Bin\right(X),$$\square )$ is a semigroup, i.e., the operation “□" as defined in general is associative. Furthermore, the left zero semigroup is an identity for this operation.

## 3. (Weakly) Implicativity in Groupoids

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Proposition**

**4.**

**Proof.**

**Theorem**

**4.**

**Proof.**

## 4. Levels of Implicativities

- Level 0:
- (i) $x\ast (y\ast x)=w\left(x\right)$ (weakly 0-implicative); (ii) $x\ast (y\ast x)=x$ (implicative).
- Level 1:
- (i) $x\ast \left(y{\square}_{1}x\right)=w\left(x\right)$ (weakly 1-implicative); (ii) $x\ast \left(y{\square}_{1}x\right)=x$ (1-implicative).
- Level i:
- (i) $x\ast \left(y{\square}_{i}x\right)=w\left(x\right)$ (weakly i-implicative); (ii) $x\ast \left(y{\square}_{i}x\right)=x$ (i-implicative).

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Proof.**

**Proposition**

**5.**

**Proof.**

**Theorem**

**7.**

**Proof.**

**Theorem**

**8.**

**Proof.**

**Corollary**

**2.**

**Proof.**

**Theorem**

**9.**

**Proof.**

## 5. Weakly Implicative Groupoids with P$\left({\mathit{L}}_{\mathit{i}}\right)$

**Proposition**

**6.**

**Proof.**

**Theorem**

**10.**

**Proof.**

**Example**

**2.**

**Theorem**

**11.**

**Proof.**

**Theorem**

**12.**

**Proof.**

**Theorem**

**13.**

- (i)
- $x\ast y=x$,
- (ii)
- $x\ast y=y$,
- (iii)
- $x\ast y=A-y$.

**Proof.**

**Theorem**

**14.**

- (i)
- $x\ast y=A$,
- (ii)
- $x\ast y=x$,
- (iii)
- $x\ast y=\frac{1}{2}(x+y)$,
- (iv)
- $x\ast y=A-\frac{1}{2}(x-y)$.

**Proof.**

## 6. Conclusions

## 7. Future Research

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Hwang, I.H.; Kim, H.S.; Neggers, J. Some Implicativities for Groupoids and *BCK*-Algebras. *Mathematics* **2019**, *7*, 973.
https://doi.org/10.3390/math7100973

**AMA Style**

Hwang IH, Kim HS, Neggers J. Some Implicativities for Groupoids and *BCK*-Algebras. *Mathematics*. 2019; 7(10):973.
https://doi.org/10.3390/math7100973

**Chicago/Turabian Style**

Hwang, In Ho, Hee Sik Kim, and Joseph Neggers. 2019. "Some Implicativities for Groupoids and *BCK*-Algebras" *Mathematics* 7, no. 10: 973.
https://doi.org/10.3390/math7100973