## 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 $h(0)=0$

### Examples

Example 1:

### Basic results

### Properties

Classtype | quasivariety ^{2)} |
---|---|

Equational theory | |

Quasiequational theory | |

First-order theory | undecidable |

Locally finite | no |

Residual size | unbounded |

Congruence distributive | no |

Congruence modular | no |

Congruence n-permutable | no |

Congruence regular | no |

Congruence uniform | no |

Congruence extension property | no |

Definable principal congruences | no |

Equationally def. pr. cong. | no |

Amalgamation property | yes |

Strong amalgamation property | yes ^{3)} |

Epimorphisms are surjective |

### Finite members

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

### Subclasses

### Superclasses

### References

^{1)}A. N. Prior,

**, Second edition, Clarendon Press, Oxford, 1962, p.316 MRreview [(Wronski1983> Andrzej Wronski,**

*Formal logic***, Math. Japon.,**

*BCK-algebras do not form a variety***28**, 1983, 211–213 MRreview [(Wronski1984> Andrzej Wronski,

**, Math. Japon.,**

*Interpolation and amalgamation properties of BCK-algebras***29**, 1984, 115–121 MRreview

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