Commutative BCK-algebras

Abbreviation: ComBCK

Definition

A commutative 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,
(4): x·y = y·x = 0 ⇒ x = y, and
(5): x·(x·y) = y·(y·x).

Remark: Note that the commutativity does not refer to the operation ·, but rather to the term operation x∧y = x·(x·y), which turns out to be a meet with respect to the following partial order:

x ≤ y ⇔ x·y = 0, with 0 as least element.

Definition

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

Remark:

Morphisms

Let A and B be commutative 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	variety
Equational theory
Quasiequational theory
First-order theory
Locally finite	No
Residual size	Unbounded
Congruence distributive	Yes
Congruence modular	Yes
Congruence n-permutable	Yes, n=3
Congruence regular
Congruence uniform
Congruence extension property
Definable principal congruences	No
Equationally definable principal congruences	No
Amalgamation property
Strong amalgamation property
Epimorphisms are surjective

Finite members

[Size 1]?: 1
[Size 2]?:
[Size 3]?:
[Size 4]?:
[Size 5]?:
[Size 6]?:

Subclasses

[Tarski algebras]?

Superclasses

BCK-algebras