A *BCK-algebra* is a structure ** A** = (

(1): ((*x*·*y*)·(*x*·*z*))·(*z*·*y*) = 0,

(2): *x*·0 = *x*,

(3): 0·*x* = 0, and

(4): *x*·*y* = *y*·*x* = 0 ⇒ *x* = *y*.

**Remark**:
*x* ≤ *y* ⇔ *x*·*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
[A. N. Prior,
*Formal logic*,
Second edition
Clarendon Press, Oxford
(1962)
MRreview], p. 316.

A *BCK-algebra* is a BCI-algebra
** A** = (

**Remark**:

Let ** A** and

Classtype | Quasivariety
[Andrzej Wroński,
BCK-algebras do not form a variety,
Math. Japon.28
(1983)
211--213
MRreview] |

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 definable principal congruences | No |

Amalgamation property | Yes |

Strong amalgamation property | Yes
[Andrzej Wronski,
Interpolation and amalgamation properties of BCK-algebras,
Math. Japon.29
(1984)
115--121
MRreview] |

Epimorphisms are surjective |

[Size 2]?:

[Size 3]?:

[Size 4]?:

[Size 5]?:

Size 6: