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.
A BCK-algebra is a BCI-algebra A = (A,·,0) such that x·0 = x.
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.
|Classtype||Quasivariety [Andrzej Wroński, BCK-algebras do not form a variety, Math. Japon. 28 (1983) 211--213 MRreview]|
|Congruence extension property||No|
|Definable principal congruences||No|
|Equationally definable principal congruences||No|
|Strong amalgamation property||Yes [Andrzej Wronski, Interpolation and amalgamation properties of BCK-algebras, Math. Japon. 29 (1984) 115--121 MRreview]|
|Epimorphisms are surjective|