Suppose you have a complete lattice $(A, \preceq)$ and a sublattice $(B, \sqsubseteq)$. By definition finite joins and meets are the same in $A$ and $B$. I wounder how infinite joins and meets relate between both lattices.
$B$ does not need to be complete, but if the join respect to $A$ of a subset of $B$ is in $B$ then it is also the join respect to $B$. My question is whether the converse holds, ie. if the join respect to $A$ is not in $B$ then the join does not exists in $B$.
The answer is no. Take $A$ to be the real interval $[0, 2]$ under the usual order and the ordinary suprema and infima. Then $A$ is a complete lattice. Let $B = {[0,1[} \cup \{2\}$ and let $S = {[0, 1[}$. Then the join (supremum) of $S$ in $A$ is $1$ and it does not belong to $B$. However, $S$ has a supremum in $B$, namely $2$.