The Strong Independence Theorem for Automorphism Groups and Congruence Lattices of Finite Lattices Theorem. Let L C and L a Be Nite Lattices, L C \ L a = F0g. Then There Exists

The Independence Theorem for the congruence lattice and the auto-morphism group of a nite lattice was proved by V. A. Baranski and A. Urquhart. Both proofs utilize the characterization theorem of congruence lattices of nite lattices (as nite distributive lattices) and the characterization theorem of auto-morphism groups of nite lattices (as nite groups). In this paper, we introduce a new, stronger form of independence. Let L be a nite lattice. A nite lattice K is a congruence-preserving extension of L, if K is an extension and every congruence of L has exactly one extension to K. Of course, then the congruence lattice of L is isomorphic to the congruence lattice of K. A nite lattice K is an automorphism-preserving extension of L, if K is an extension and every automorphism of L has exactly one extension to K, and in addition, every automorphism of K is the extension of an automorphism of L. Of course, then the automorphism group of L is isomorphic to the automorphism group of K. a nite atomistic lattice K that is a congruence-preserving extension of L C and an automorphism-preserving extension of L A. In fact, both extensions preserve the zero. Of course, the congruence lattice of K is isomorphic to congruence lattice of L C , and the automorphism group of K is isomorphic to the automorphism group of L A .