Table of Contents
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 $h(1)=1$.
Examples
Example 1:
Basic results
Properties
Finite members
$\begin{array}{lr}
f(1)= &1
f(2)= &
f(3)= &
f(4)= &
f(5)= &
f(6)= &
\end{array}$
