Let ${\cal L}$ be defined as in [this question](https://mathoverflow.net/questions/215582/how-big-is-the-lattice-of-all-functions). Is there a surjective lattice homomorphism $f: {\cal L}\to \mathbb{N}^\mathbb{N}$, where $\mathbb{N}^\mathbb{N}$ is the set of all functions, ordered point-wise?



Is there a surjective lattice homomorphism $f: {\cal L}\to \mathbb{N}^\mathbb{N}$?
$\mathcal L$ has a cofinal $\omega$-chain and $\mathbb N^{\mathbb N}$ does not.


