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If $L$ is a complete lattice and $P$ is a poset and $f: L\to P$ is an order preserving surjective map, does this imply that $P$ is a (complete) lattice?
If $L$ is a complete lattice and $P$ is a poset and $f: L\to P$ is an order preserving map, does this imply that $P$ is a (complete) lattice?
If $L$ is a complete lattice and $P$ is a poset and $f: L\to P$ is an order preserving surjective map, does this imply that $P$ is a (complete) lattice?
