Example of closure operator on a lattice such that $C(A \cup B) \neq C(A) \cup C(B) $

Question

I'm looking for this example so that $L_C$ is not a sublattice. I discard topological closure and closure given by $C(A)$ =subgroup generated by $A$ and can't think of any other example.

Answer 1

Start with the lattice $L$ of all subsets of $\{1,2,3\}$. Define $C(A)=\{1,2,3\}$ for all $A\in L$ of size $2$ and $C(A)=A$ for all other $A\in L$.