BCK-join-semilattices

Abbreviation: BCKJSlat

Definition

A \emph{BCK-join-semilattice} is a structure $\mathbf{A}=\langle A,\vee,\rightarrow,1\rangle$ of type $\langle 2,2,0\rangle$ such that

(1): $(x\rightarrow y)\rightarrow ((y\rightarrow z)\rightarrow (x\rightarrow z)) = 1$

(2): $1\rightarrow x = x$

(3): $x\rightarrow 1 = 1$

(4): $x\rightarrow (x\vee y) = 1$

(5): $x\vee((x\rightarrow y)\rightarrow y) = ((x\rightarrow y)\rightarrow y)$

$\vee$ is idempotent: $x\vee x = x$

$\vee$ is commutative: $x\vee y = y\vee x$

$\vee$ is associative: $(x\vee y)\vee z = x\vee (y\vee z)$

Remark: $x\le y \iff x\rightarrow y=1$ is a partial order, with $1$ as greatest element, and $\vee$ is a join for this order. 1)

Morphisms

Let $\mathbf{A}$ and $\mathbf{B}$ be BCK-join-semilattices. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:

$h(x\vee y)=h(x)\vee h(y)$, $h(x\rightarrow y)=h(x)\rightarrow h(y)$ and $h(1)=1$

Example 1:

Properties

Classtype variety

Finite members

$\begin{array}{lr} f(1)= &1 f(2)= & f(3)= & f(4)= & f(5)= & f(6)= & \end{array}$

References

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