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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?

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up vote 4 down vote accepted

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

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