BCK-algebras

Abbreviation: BCK

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): 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.

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 : A→B that is a homomorphism: h(x·y) = h(x)·h(y) and h(0) = 0.

Some results

Examples

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

Finite members

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Subclasses

commutative BCK-algebras

Superclasses

BCI-algebras