The Strong Independence Theorem for automorphism groups and congruence lattices of arbitrary lattices

George Grätzer (1) , Friedrich Wehrung (2)

1 Department of Mathematics
2 LMNO - Laboratoire de Mathématiques Nicolas Oresme

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.

Mots clés: box product; tensor product; gluing; direct limit; Lattice; automorphism; congruence

Domaines: Mathématiques générales [math.GM]

George Grätzer, Friedrich Wehrung. The Strong Independence Theorem for automorphism groups and congruence lattices of arbitrary lattices. Advances in Applied Mathematics, 2000, 24 (3), pp.181-221. ⟨10.1006/aama.1999.0661⟩. ⟨hal-00004030⟩