## BCK-lattices

Abbreviation: BCKlat

### Definition

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

$\langle A,\vee,\rightarrow,1\rangle$ is a BCK-join-semilattice

$\langle A,\wedge,\rightarrow,1\rangle$ is a BCK-meet-semilattice

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

##### Morphisms

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

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

Example 1:

### Properties

Classtype variety yes yes yes $n=2$

### Finite members

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

### References

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