BCK-meet-semilattices

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Difference (from revision 1 to current revision) (minor diff)

Changed: 18,19c18,36
 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. [ Pawel M. Idziak, Lattice operation in BCK-algebras, Math. Japon. 29 (1984) 839--846 MRreview ]

Changed: 24c41
 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.

Definition

A BCK-meet-semilattice is a structure A = (A,∧,→,1) of type (2,2,0) such that

(1):   (xy)→((yz)→(xz))  = 1
(2):   1→x  = x
(3):   x→1  = 1
(4):   (xy)→y  = 1
(5):   x∧((xy)→y)  = x
is idempotent:   xx  = x
is commutative:   xy  = yx
is associative:   (xy)∧z  = x∧(yz)

Remark: x ≤ y   ⇔   xt 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 : AB that is a homomorphism: h(xy) = h(x)∧h(y) and h(xy) = h(x)→h(y) and h(1) = 1.

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BCK-lattices

Superclasses

BCK-algebras

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