## BCK-algebras

Abbreviation: BCK

### Definition

A 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).

### Definition

A BCK-algebra is a BCI-algebra $\mathbf{A}=\langle A,\cdot ,0\rangle$ such that

$x\cdot 0 = x$

##### Morphisms

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:

### Properties

Classtype quasivariety 2) undecidable no unbounded no no no no no no no no yes yes 3)

### Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &3\\ f(4)= &14\\ f(5)= &88\\ f(6)= &775\\ \end{array}$

### References

1) A. N. Prior, Formal logic, Second edition, Clarendon Press, Oxford, 1962, p.316
2) Andrzej Wronski,BCK-algebras do not form a variety, Math. Japon., 28, 1983, 211–213
3) Andrzej Wronski,Interpolation and amalgamation properties of BCK-algebras, Math. Japon., 29, 1984, 115–121