What are ideals in a complete lattice?

Question

If I have a complete lattice $L$, what conditions do I need for $I\subseteq L$ to be an ideal?

In a general lattice the conditions are:

• $I$ is a lower set;
• $I$ is closed under (finite) joins.

For a complete lattice, do we require that $I$ be closed under arbitrary joins? Or is there a special name for such a "better" ideal?

Answer 1

OK, I've found the answer here. Such a "better" ideal is called a complete ideal.