BCK-meet-semilattices

Abbreviation: BCKJMlat

Definition

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]

Morphisms

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.

Some results

Examples

Properties

Finite members

Subclasses

Superclasses