Given a poset $(P, \leq)$, is there a complete Boolean lattice $B$ and an order-preserving map $i\_P: P\to B$ such that for any complete Boolean lattice $B'$ and order-preserving map $f: P\to B'$ there is an order-preserving map $g: B\to B'$ such that $f = g\circ i\_P$?



Boolean completion of a partially ordered set
The answer is yes. Take $B := 2^P$ to be the power set of $P$, and $i\_P(x)$ to be the principal down-set generated by $x$. Now for $A\subseteq P$, put $g(A) := \sup f(A)$. This makes $f = g\circ i\_P$ hold by construction.


However, I don't think that this is a universal property, since the $g$ may not be unique. In fact, using maps which preserve all joins and suprema as the usual morphisms of complete Boolean algebras, the existence of a universal complete Boolean algebra generated by $P$ is not guaranteed, and fails even for [$P$ being the countable antichain](https://thesis.library.caltech.edu/6561/1/Hales_aw_1962.pdf).


