Name for partial order that is $\omega$-complete and co-$\omega$-complete?

Question

Is every $\omega$-complete (every countable ascending chain has a join) and co-$\omega$-complete (every countable descending chain has a meet) partial order already a complete lattice? My guess is that this is not the case, as an $\omega$-complete partial order does not need to be an DCPO. So, do these structures have a special name?

Answer 1

Such a partial order does not even have to be a lattice. For instance, take any finite poset that is not a lattice. Every ascending or descending chain is eventually constant, and so has a join or meet.

Answer 2

No, of course not.

Consider the partial order $(\mathcal P_{\omega_1}(\Bbb R),\subseteq)$, that is all the countable subsets of $\Bbb R$ ordered by inclusion.

Every countable chain has a supremum: its union; and an infimum: its intersection. Indeed, every countable subset has a join and a meet.

But this is most certainly not a complete lattice since it has no maximum.