Topological representation of some lattices

In this paper, we first give a topological representation of some algebraic lattices of ideals of C(X). Next, we apply these results and prove that a space X is normal if and only if the lattice of closed fixed ideals of C(X) is a sublattice of the lattice of ideals of C(X). It is proved that if two rings C(X) and C(Y) are isomorphic, then two lattices Z ∘[X] and Z ∘[Y] are isomorphic. We conclude that two rings C *(X) and C *(Y) are isomorphic if and only if two lattices Z[βX] and Z[βY] are isomorphic.