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edited May 19, 2015 at 11:49

Dominic van der Zypen

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Dedekind-MacNeille completion of $\omega^\omega$ and a subsetthe strictly increasing members of that poset$\omega^\omega$

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asked May 19, 2015 at 11:40

Dominic van der Zypen

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Dedekind-MacNeille completion of $\omega^\omega$ and a subset of that poset

Let $\omega^\omega$ denote the set of all functions $f:\omega\to\omega$ ordered by $f\leq g$ iff $f(n) \leq g(n)$ for all $n\in \omega$.

Set $K = \{f\in \omega^\omega: m<n\in \omega \implies f(m)<f(n)\}$. If $\textbf{DM}(\cdot)$ denotes the Dedekind-MacNeille completion, do we have $\textbf{DM}(\omega^\omega) \cong \textbf{DM}(K)$?

order-theory lattice-theory