# Is there a surjective lattice homomorphism $f: {\cal L}\to \mathbb{N}^\mathbb{N}$?

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 point-wise?

• 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 $f_k(n) = n+k$, $k=1, 2, 3, \ldots$.) – Keith Kearnes Aug 26 '15 at 7:26
• Oh - right -- excellent argument! Can you quickly put this in an answer? – Dominic van der Zypen Aug 26 '15 at 7:33

$\mathcal L$ has a cofinal $\omega$-chain and $\mathbb N^{\mathbb N}$ does not.