Complete Congruence Lattices of Complete Distributive Lattices

✍ Scribed by G. Gratzer; E.T. Schmidt

• Publisher: Elsevier Science
• Year: 1995
• Tongue: English
• Weight: 984 KB
• Volume: 171
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1995.1009

✦ Synopsis

In 1983. Wille raised the question: Is elery complete lattice (L) isomorphic to the lattice of complete congruence relations of a suitable complete lattice K? In 1988 . this was answered in the affirmative by the first author. A number of papers have been published on this problem by Freese, Johnson, Lakser, Teo, and the authors. In the present paper we prove that (K) can always be chosen as a complete distributive lattice. In fact, we prove the following more general result:

Tireorem. Let in be a regular cardinal (>\mathbf{N}_{0}). Eiery m-algetraic lattice (L) can be represented as the lattice of it-complete congruence relations of an (\mathrm{tt}-) complete distributitice lattice (K . w 1995) Academic Press. Inc.