What is an example of two bounded lattices $L, K$ such that there exist surjective lattice homomorphisms $f:L\to K$ and $g:K\to L$, but there are no injective lattice homomorphisms between $L, K$?



Bounded lattices with lattice surjections but no injections between them
Let $\bf n$ be the $n$-element antichain. Define $L\_{m,n}$ to be the lattice that is the ordinal sum ${\bf 1}+{\bf m}+{\bf 1}+{\bf n}+{\bf 1}+{\bf m}+{\bf 1}+{\bf n}+\cdots$, with $\omega$-many summands. Let $L\_{m,n}^\*$ be the bounded lattice obtained by adding a top element $1$ to $L\_{m,n}$ and letting the original bottom element of $L\_{m,n}$ be $0$. It is not too hard to see that $L\_{3,4}^\*$ and $L\_{4,3}^\*$ are bounded lattices, each has a surjective homomorphism onto the other, and neither has an injective ($0$-preserving) homomorphism to the other.


