BCK-join-semilattices

Abbreviation: BCKJSlat

Definition

A BCK-join-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→(x∨y)  = 1
(5):   x∨((x→y)→y)  = ((x→y)→y)
∨ is idempotent:   x∨x  = x
∨ is commutative:   x∨y  = y∨x
∨ is associative:   (x∨y)∨z  = x∨(y∨z)

Remark: x ≤ y   ⇔   x→y = 1 is a partial order, with 1 as greatest element, and ∨ is a join for this order.

[Pawel M. Idziak, Lattice operation in BCK-algebras, Math. Japon. 29 (1984) 839--846 MRreview]

Morphisms

Let A and B be BCK-join-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.

Some results

Examples

Properties

Finite members

Subclasses

Superclasses