lattice theory questions about ideals

Question

Let $L$ be a (bounded) lattice

a) Give an example of an ideal in $L$ that is not a normal ideal. (A normal ideal is the element in $DM(L)$, Dedekind-MacNeille completion

This question I think about an infinite lattice L, but I don't know which one will work.

b) Is it true that if $L$ is complete, then $L$ is isomorphic to $\mathbb{I}\left(L\right)$, where $\mathbb{I}\left(L\right)$ is the set of all ideals of $L$?

I think it is not true, but quite stuck.

Any hint is appreciated. Thank you.

Answer 1

An example of a lattice with a non-normal ideal is the following:

Here, if $I = \{0,a_1,a_2,\ldots\} = L\setminus\{b,1\}$, then $I^{ul}=L$, because $1$ is the only upper bound of $I$.

In this answer there is an example of a complete lattice $\mathbf L$ that is not isomorphic to its ideal lattice (in that case, labelled $\mathbf K$).