Non-principal Ideals in a Complete Lattice

Question

Given a complete lattice is it possible to have orderideals which are not principal? Can one not always just join together every element of the ideal to get its maximal, generating element? What about for frames?

Thanks!

Answer 1

Short answer is no. In order to get a counterexample, consider the Boolean algebra of subsets of the natural numbers $\mathcal P(\mathbb N)$ and let $FIN$ denote the ideal of finite subsets of $\mathbb N$. Observe that $\mathcal P(\mathbb N)$ is a complete lattice and $FIN$ is a not principal ideal.