A uniform refinement property for congruence lattices
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Abstract:
The Congruence Lattice Problem asks whether every algebraic distributive lattice is isomorphic to the congruence lattice of a lattice. It was hoped that a positive solution would follow from E. T. Schmidt’s construction or from the approach of P. Pudlák, M. Tischendorf, and J. Tůma. In a previous paper, we constructed a distributive algebraic lattice $A$ with $\aleph _2$ compact elements that cannot be obtained by Schmidt’s construction. In this paper, we show that the same lattice $A$ cannot be obtained using the Pudlák, Tischendorf, Tůma approach. The basic idea is that every congruence lattice arising from either method satisfies the Uniform Refinement Property, that is not satisfied by our example. This yields, in turn, corresponding negative results about congruence lattices of sectionally complemented lattices and two-sided ideals of von Neumann regular rings.References
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Additional Information
- Friedrich Wehrung
- Affiliation: Département de Mathématiques, Université de Caen, 14032 Caen Cedex, France
- MR Author ID: 242737
- Email: gremlin@math.unicaen.fr
- Received by editor(s): September 20, 1996
- Received by editor(s) in revised form: February 13, 1997, and May 30, 1997
- Communicated by: Ken Goodearl
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 127 (1999), 363-370
- MSC (1991): Primary 06A12, 06B10; Secondary 16E50
- DOI: https://doi.org/10.1090/S0002-9939-99-04558-X
- MathSciNet review: 1468207