$( A,\le)$ and $(A',\le')$ are partially ordered sets. A map $\phi : A \to A'$ is called order preserving from $(A, \le)$ to $(A', \le')$ if for all $x, y \in A : x \le y \implies \phi(x) \le' \phi(y)$
For example:
For $(P(\{1,2\}), \subseteq)$ and $(\mathbb{N}, \le)$ is the map $\phi : P(\{1,2\}) \to \mathbb{N} : X \mapsto \sum_{n \in X} n\:$ order preserving from $(P(\{1,2\},\subseteq)$ to $(\mathbb{N}, \le)$ - and also injective.
1.) Is there injective, order preserving map from $(\mathbb{N},\le)$ to $(\mathbb{N}, | )$??
2.)Is there injective, order preserving map from $(\mathbb{N_+}, | )$ to $(\mathbb{N},\le)$?
For 1: $$ \phi(n) = 2^n $$
For 2: $$ \phi(n)=n. $$