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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): (xy)((yz)(xz))=1

(2): 1x=x

(3): x1=1

(4): x(xy)=1

(5): x((xy)y)=((xy)y)

is idempotent: xx=x

is commutative: xy=yx

is associative: (xy)z=x(yz)

Remark: xyxy=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:AB that is a homomorphism:

h(xy)=h(x)h(y), h(xy)=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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1), 2) Pawel M. Idziak, \emph{Lattice operation in BCK-algebras}, Math. Japon., \textbf{29}, 1984, 839–846 MRreview

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