A closure operator on a set $A$ is a function $C: \mathcal{P}(A) \to \mathcal{P}(A)$ satisfying following axioms:
- $X ⊆ Y \implies C(X) ⊆ C(Y)$
- $X ⊆ C(X)$
It may also satisfy some additional axioms:
- $C(C(X)) = C(X)$
- $C(∅) = ∅$
- $C(\{x\}) = \{x\}$ for all $x ∈ A$
We call a set $X ⊆ A$ closed (with respect to $C$) if $C(X) = X$. To every closure operator $C$ we may assign the set of all closed sets $F(C)$, which is a complete lattice. On the other hand to every complete lattice $L$ we can assign closure operator $G(L)$ on the set of all atoms $At(L)$: $G(L)(X) := \{x ∈ At(L): x ≤ \bigvee X\}$.
$F$ and $G$ gives natural correspondence between closure operators satisfying the additional axioms and atomistic complete lattices: $G(F(C)) \cong C$ and $F(G(L)) \cong L$. (A lattice is atomistic if every element is a join of atoms.)
My question is following:
Can this natural correspondence be extended? Is there a bigger class of complete lattices and $G'$ which maps such lattice to a closure operator such that $F$, $G'$ form a correspondence exteding $F$, $G$?