In the book, General Lattice Theory, 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, in the affirmative, by V.A. Baranskii and A. Urquhart in 1978, independently. In 1995, the first author and E.T. Schmidt proved a much stronger result, the strong independence ofthe 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.
@article{hal-00004030,
author = {Gr\"atzer, George and Wehrung, Friedrich},
title = {The Strong Independence Theorem for automorphism groups and congruence lattices of arbitrary lattices},
journal = {HAL},
volume = {2000},
number = {0},
year = {2000},
language = {en},
url = {http://dml.mathdoc.fr/item/hal-00004030}
}
Grätzer, George; Wehrung, Friedrich. The Strong Independence Theorem for automorphism groups and congruence lattices of arbitrary lattices. HAL, Tome 2000 (2000) no. 0, . http://gdmltest.u-ga.fr/item/hal-00004030/