Order embedding from a poset into a complete lattice Let $\mathfrak{A}$ is an arbitrary poset.
Does it necessarily exist an order embedding from $\mathfrak{A}$ into some complete lattice $\mathfrak{B}$, which preserves all suprema and infima defined in $\mathfrak{A}$?
 A: The answer is yes. 
First, given any partially ordered set $P$, there is always a complete lattice that contains $P$, for example, the collection of downward closed subsets of $P$, under containment, where $p\in P$ is identified with $\{x\in P\mid x\le p\}$. 
But, in fact, any poset $\mathcal P=(P,\le)$ admits a minimal completion $\mathcal L$, in the sense that $\mathcal L$ is a complete lattice containing $\mathcal P$, $\mathcal L$ embeds (as a poset) into any complete lattice that contains $\mathcal P$ (with the embedding fixing $P$ pointwise), and $\mathcal L$ preserves all suprema and infima (including suprema and infima of infinite subsets) already present in $P$.
As with the rationals and the reals, a natural way of defining this minimal completion $\mathcal L$ of $\mathcal P$ is simply to take as $\mathcal L$ the collection of all cuts in $\mathcal P$, partially ordered by saying that $(A,B)\le(C,D)$ iff $A\subseteq C$. 
Here, a cut of $\mathcal P$ is a pair $(A,B)$ of subsets of $P$ such that $B$ is the collection of upper bounds of $A$, and $A$ is the collection of lower bounds of $B$.
This construction, that generalizes Dedekind's construction of the reals as cuts of rationals, is called the Dedekind-MacNeille (or normal) completion, first introduced in 

Holbrook M. MacNeille. Partially ordered sets, Trans. Amer. Math. Soc. 42 (3), (1937), 416–460. MR1501929,

where complete details can be found.  
