|GEORGE GRÄTZER, Department of Mathematics, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada|
|Independence Theorems for automorphism groups and congruence lattices of lattices|
In my book, General Lattice Theory, in 1978, I raised the following problem:
Let L be a 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?
Several papers have been published providing technical tools that could be used to attack this problem, in particular,
earlier papers by G. Birkhoff and R. Frucht;
more recent papers by V. A. Baranski, M. Tischendorf, and A. Urquhart;
a number of relevant categorical results by the Prague group (reported, in part, in a book of A. Pultr and V. Trnková);
a number of papers by G. Grätzer and E. T. Schmidt on congruence-preserving extensions;
earlier papers on the topic of tensor products of lattices with zero by J. Anderson and N. Kimura, G. A. Fraser, and G. Grätzer, H. Lakser, and R. W. Quackenbush and a series of very recent papers on the same topic (and on some generalizations) by G. Grätzer and F. Wehrung.
Based on these contributions, F. Wehrung and I have succeeded in solving this problem.
To state the new results, we need two definitions.
Let L be a lattice. A lattice K is a congruence-preserving extension of L, if K is an extension of L and every congruence of L extends to exactly one congruence of K. Of course, then the congruence lattice of L is isomorphic to the congruence lattice of K.
A lattice K is an automorphism-preserving extension of L, if K is an extension of L 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.
The Strong Independence Theorem for Lattices with Zero.
Let and be lattices with zero, let have more than one element. Then there exists a lattice K that is a -preserving extension of both and , an automorphism-preserving extension of , and a congruence-preserving extension of .
The Strong Independence Theorem for Lattices.
Let and be lattices, let have more than one element. Then there exists a lattice K that is an automorphism-preserving extension of and a congruence-preserving extension of .