Abbreviation: BCI

A \emph{BCI-algebra} is a structure $\mathbf{A}=\langle A,\cdot ,0\rangle$ of type $\langle 2,0\rangle$ such that

(1): $((x\cdot y)\cdot (x\cdot z))\cdot (z\cdot y) = 0$

(2): $(x\cdot (x\cdot y))\cdot y = 0$

(3): $x\cdot x = 0$

(4): $x\cdot y=y\cdot x= 0 \Longrightarrow x=y$

(5): $x\cdot 0 = 0 \Longrightarrow x=0$

Remark:

Let $\mathbf{A}$ and $\mathbf{B}$ be BCI-algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:

$h(x\cdot y)=h(x)\cdot h(y) \mbox{ and } h(0)=0$

Example 1:

$\begin{array}{lr} f(1)= &1
f(2)= &
f(3)= &
f(4)= &
f(5)= &
f(6)= &
\end{array}$

BCK-algebras

Groupoids