×

The strong independence theorem for automorphism groups and congruence lattices of finite lattices. (English) Zbl 0817.06005

Summary: The Independence Theorem for the congruence lattice and the automorphism group of a finite lattice was proved by V. A. Baranskiĭ and A. Urquhart. Both proofs utilize the characterization theorem of congruence lattices of finite lattices (as finite distributive lattices) and the characterization theorem of automorphism groups of finite lattices (as finite groups). We introduce a new, stronger form of independence.
Let \(L\) be a finite lattice. A finite 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 finite 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 \(K\).
Theorem. Let \(L_ C\) and \(L_ A\) be finite lattices, \(L_ C \cap L_ A = \{0\}\). Then there exists a finite 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 the congruence lattice of \(L_ C\), and the automorphism group of \(K\) is isomorphic to the automorphism group of \(L_ A\).

MSC:

06B05 Structure theory of lattices
06B10 Lattice ideals, congruence relations
PDFBibTeX XMLCite
Full Text: EuDML EMIS