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 $L, K$?
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asked Apr 10, 2017 at 7:51
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2 Answers
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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.
answered Apr 18, 2017 at 5:02
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I think you can remove the assumption of 0-preservation in @KeithKearnes' answer by replacing $\mathbf 3$, $\mathbf 4$, and the 0 element of each lattice by three mutually non-embeddable bounded lattices, thereby obtaining an answer to the question as originally stated.
answered Apr 18, 2017 at 6:09