For $A, B \subseteq \omega$ we set $A \leq\_{\text{inj}} B$ if there is an injective and order-preserving map $f:\omega\to \omega$ , such that $f(A)$ is a down-set of $B$. It is easy to see that $\leq\_{\text{inj}}$ is an ordering relation on ${\cal P}(\omega)$. 


Is $({\cal P}(\omega), \leq\_{\text{inj}})$ a lattice? Is it [distributive](https://en.wikipedia.org/wiki/Distributive_lattice)?



Is $({\cal P}(\omega), \leq_{\text{inj}})$ a distributive lattice?
Let
$$A=\{a\_0<a\_0+a\_1<a\_0+a\_1+a\_2<\dots\},\qquad B=\{b\_0<b\_0+b\_1<\dots\},$$
so that the $a\_i$ and $b\_i$ are the gaps in $A$ and $B$.
Then $A\le\_{\mathrm{inj}}B$ iff $a\_i\le b\_i$ for each $i$. Thus $(\mathcal P(\omega),\le\_{\mathrm{inj}})$ is indeed a distributive lattice, isomorphic to $\omega^\omega$ with the product order.


