# BCK-algebras

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### Definition

A BCK-algebra is a structure A = (A,·,0) of type (2,0) such that

(1):   ((x·y)·(x·z))·(z·y)  = 0,
(2):   x·0  = x,
(3):   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.

### Definition

A BCK-algebra is a BCI-algebra A = (A,·,0) such that x·0  = x.

Remark:

### Morphisms

Let A and B be BCK-algebras. A morphism from A to B is a function h : AB that is a homomorphism: h(x·y) = h(xh(y) and h(0) = 0.

### Properties

 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

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### Subclasses

commutative BCK-algebras

### Superclasses

BCI-algebras

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