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

DOI:
https://doi.org/10.1090/S0002-9939-99-04558-X

MathSciNet review:
1468207

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Abstract | References | Similar Articles | Additional Information

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 with compact elements that cannot be obtained by Schmidt's construction. In this paper, we show that the same lattice 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

Email:
gremlin@math.unicaen.fr

DOI:
https://doi.org/10.1090/S0002-9939-99-04558-X

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