Let $\kappa$ a regular uncountable cardinal and $\mathbb{P}$ and $\mathbb{Q}$ poset.
$\mathbb{P}$ has the $\kappa$-Knaster property iff for every $A \subseteq \mathbb{P}$ of size $\kappa$ there is $B \subseteq A$ of size $\kappa$ consisting of pairwise compatible elements.
$\mathbb{P}$ has precaliber $\kappa$ if, given $Q \subseteq \mathbb{P}$ of size, there is a centered $Q^{*} \subseteq Q$ of size $\kappa$.
Questions:
$(i)$ If there is a complete embedding $i:\mathbb{P}\to \mathbb{Q}$ and $\mathbb{Q}$-Knaster then $\mathbb{P}$-Knaster.?
$(ii)$ If there is a dense embedding $i:\mathbb{P}\to \mathbb{Q}$ and $\mathbb{P}$ has precaliber $\kappa$ then $\mathbb{Q}$ has precaliber.?
Thanks.