all lattice $\{0,1\}$- homomorphisms from $L$ to $L \times K$

Question

let K be the lattice of the chain $\{0,1\}$ and $L$ be the lattice bellow

I want to know all $\{0,1\}$-homomorphisms (homomorphism taking zero into zero and unit into unit) from $L$ to $L \times K$.

I guess there is not any but i cant prove. Can you help please?

Answer 1

Notice that $L \times K$ is the cube, aka the eight-element Boolean algebra, or the power-set algebra on a three-element set.
See a picture of it here.

So, if you have the images of $0$ and $1$ fixed, and given that $a \vee b = 1$ and $a \wedge b = 0$, the image of one of these elements determines the image of the other. Indeed, the image of $a$ and of $b$ must always be the (unique) complement of each other in $L \times K$.

So you have eight choices for $h(a)$, which determines also the image of $b$, so eight $\{0,1\}$-homomorphisms.
Notice that six of these are one-to-one.