When is a join semilattice not a meet semilattice?

Suppose $\Omega$ is your favourite set of more than two elements. The collection $\mathcal P_+(\Omega)$ of nonempty subsets of $\Omega$ becomes a poset when ordered by set inclusion: For any $A,B \subset \Omega$ we write $A \le B$ to mean $A \subset B$. Moreover $(\mathcal P_+(\Omega), \le)$ is a join-semilattice with the join given by $A \vee B = A \cup B$. But this is not a meet-semilattice because if $A$ and $B$ are disjoint they have no shared lowed bound at all.

Now define $\mathcal P_-(\Omega)$ as the collection of proper subsets of $\Omega$. Then $(\mathcal P_-(\Omega), \le)$ is a meet-semilattice with meet given by $A \wedge B = A \cap B$. But again if $A \cup B = \Omega$ then $A$ and $B$ have no shared upper bound at all. So this cannot be a join-semilattice.

You should prove for both examples if it is not obvious. Keep in mind that $\varnothing \notin \mathcal P_+(\Omega)$ is insufficient, as maybe some other element of $\mathcal P_+(\Omega)$ acts as the lower bound of $A$ and $B$.

Question: Can you choose some more sophisticated family $\mathcal Q \subset \mathcal P(\Omega)$ so the resulting poset $(\mathcal Q, \le)$ is a doubly-directed join-but-not-meet-semilattice?