The Strong Independence Theorem for Automorphism Groups and Congruence Lattices of Arbitrary Lattices

AbstractIn the book, General Lattice Theory (Birkhäuser, Basel, 1978), the first author raised the following problem (Problem II.18):Let L be a nontrivial lattice and let G be a group. Does there exist a lattice K such that K and L have isomorphic congruence lattices and the automorphism group of K is isomorphic to G?The finite case was solved, independently, in the affirmative, by V. A. Baranskiĭ (1979, in “Abstracts of Lectures of the 15th All-Soviet Algebraic Conference, Krasnojarsk, July 1979,” Vol. 1, p. 11) and A. Urquhart (1978, Algebra Universalis8, 45–58). In 1995, the first author and E. T. Schmidt (Beitrage Algebra Geom.36, 97–108) proved a much stronger result, the strong independence of the automorphism group and the congruence lattice in the finite case. In this paper, we provide a full affirmative solution of the above problem. In fact, we prove much stronger results, verifying strong independence for general lattices and also for lattices with zero