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BCK-join-semilattices
Abbreviation: BCKJSlat
Definition
A \emph{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. 1)
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), h(x→y)=h(x)→h(y) and h(1)=1
Examples
Example 1:
Basic results
Properties
Finite members
f(1)=1f(2)=f(3)=f(4)=f(5)=f(6)=
Subclasses
Superclasses
References
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