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A uniform refinement property
for congruence lattices

Author: Friedrich Wehrung
Journal: Proc. Amer. Math. Soc. 127 (1999), 363-370
MSC (1991): Primary 06A12, 06B10; Secondary 16E50
MathSciNet review: 1468207
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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. Tuma. 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, Tuma 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.

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Additional Information

Friedrich Wehrung
Affiliation: Département de Mathématiques, Université de Caen, 14032 Caen Cedex, France

Keywords: Semilattices, weakly distributive homomorphisms, congruence splitting lattices, uniform refinement property, von Neumann regular rings
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
Article copyright: © Copyright 1999 American Mathematical Society