x ≤ y ⇔ x·y = 0 is a partial order, with 0 as least element, and ∧|
is a meet in this partial order.
x ≤ y ⇔ x→t y = 1 is a partial order, with 1 as greatest element, and ∧ is a meet in this partial order.|
h(x∧y) = h(x)∧h(y) and h(x·y) = h(x)·h(y) and h(0) = 0.|
h(x∧y) = h(x)∧h(y) and h(x→y) = h(x)→h(y) and h(1) = 1.|
A BCK-meet-semilattice is a structure A = (A,∧,→,1) of type (2,2,0) such that
(1): (x→y)→((y→z)→(x→z)) = 1
(2): 1→x = x
(3): x→1 = 1
(4): (x∧y)→y = 1
(5): x∧((x→y)→y) = x
∧ is idempotent: x∧x = x
∧ is commutative: x∧y = y∧x
∧ is associative: (x∧y)∧z = x∧(y∧z)
Remark: x ≤ y ⇔ x→t y = 1 is a partial order, with 1 as greatest element, and ∧ is a meet in this partial order.
[Pawel M. Idziak, Lattice operation in BCK-algebras, Math. Japon. 29 (1984) 839--846 MRreview]
Let A and B be BCK-meet-semilattices. 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(x→y) = h(x)→h(y) and h(1) = 1.
|Congruence n-permutable||yes, n = 2|
|Congruence extension property|
|Definable principal congruences|
|Equationally definable principal congruences|
|Strong amalgamation property|
|Epimorphisms are surjective|