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?
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asked Aug 26, 2015 at 6:51
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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 KearnesCommented Aug 26, 2015 at 7:26
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$\begingroup$ Oh - right -- excellent argument! Can you quickly put this in an answer? $\endgroup$– Dominic van der ZypenCommented Aug 26, 2015 at 7:33
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1 Answer
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$\mathcal L$ has a cofinal $\omega$-chain and $\mathbb N^{\mathbb N}$ does not.
answered Aug 26, 2015 at 7:35