Let ${\cal L}$ be defined as in this question. 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 pointwise?
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$\begingroup$ I think $\mathcal L$ has a cofinal $\omega$chain and $\mathbb N^{\mathbb N}$ does not. (For the chain in $\mathcal L$, consider the sequence $f_k(n) = n+k$, $k=1, 2, 3, \ldots$.) $\endgroup$ – Keith Kearnes Aug 26 '15 at 7:26

$\begingroup$ Oh  right  excellent argument! Can you quickly put this in an answer? $\endgroup$ – Dominic van der Zypen Aug 26 '15 at 7:33
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$\mathcal L$ has a cofinal $\omega$chain and $\mathbb N^{\mathbb N}$ does not.