Abbreviation: BCK
A \emph{BCK-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): $x\cdot 0 = x$
(3): $0\cdot x = 0$
(4): $x\cdot y=y\cdot x= 0 \Longrightarrow x=y$
Remark: $x\le y \iff x\cdot y=0$ is a partial order, with $0$ as least element.
BCK-algebras provide algebraic semantics for BCK-logic, named after the combinators B, C, and K by C. A. Meredith, see 1).
A \emph{BCK-algebra} is a BCI-algebra $\mathbf{A}=\langle A,\cdot ,0\rangle$ such that
$x\cdot 0 = x$
Let $\mathbf{A}$ and $\mathbf{B}$ be BCK-algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: $h(x\cdot y)=h(x)\cdot h(y)$ and $h(0)=0$
Example 1:
$\begin{array}{lr}
f(1)= &1
f(2)= &1
f(3)= &3
f(4)= &14
f(5)= &88
f(6)= &775
\end{array}$