Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

A uniform refinement property for congruence lattices

Author(s): 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 with $\aleph _2$ 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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