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A uniform refinement property for congruence lattices. (English) Zbl 0902.06006
Summary: 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.

06B10 Lattice ideals, congruence relations
06D05 Structure and representation theory of distributive lattices
16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
06A12 Semilattices
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