A uniform refinement property for congruence lattices

Wehrung, Friedrich

doi:10.1090/S0002-9939-99-04558-X

A uniform refinement property for congruence lattices
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Proc. Amer. Math. Soc. 127 (1999), 363-370 Request permission

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

P. Ara, K. R. Goodearl, E. Pardo and K. C. O’Meara, Separative cancellation for projective modules over exchange rings, Israel J. Math., to appear.
G. M. Bergman, Von Neumann regular rings with tailor-made ideal lattices, unpublished note (1986).
Garrett Birkhoff, Lattice theory, 3rd ed., American Mathematical Society Colloquium Publications, Vol. 25, American Mathematical Society, Providence, R.I., 1979. MR 598630
Hans Dobbertin, Refinement monoids, Vaught monoids, and Boolean algebras, Math. Ann. 265 (1983), no. 4, 473–487. MR 721882, DOI 10.1007/BF01455948
K. R. Goodearl, von Neumann regular rings, 2nd ed., Robert E. Krieger Publishing Co., Inc., Malabar, FL, 1991. MR 1150975
K. R. Goodearl, Von Neumann regular rings and direct sum decomposition problems, Abelian Groups and Modules, Padova 1994 (A. Facchini and C. Menini, eds.), Dordrecht (1995) Kluwer, pp. 249–255.
George Grätzer, Lattice theory. First concepts and distributive lattices, W. H. Freeman and Co., San Francisco, Calif., 1971. MR 0321817
George Grätzer, General lattice theory, Lehrbücher und Monographien aus dem Gebiete der Exakten Wissenschaften, Mathematische Reihe, Band 52, Birkhäuser Verlag, Basel-Stuttgart, 1978. MR 504338
G. Grätzer and E.T. Schmidt, Congruence lattices of lattices, Appendix C in G. Grätzer, General Lattice Theory, Second Edition, to appear.
John von Neumann, Continuous geometry, Princeton Mathematical Series, No. 25, Princeton University Press, Princeton, N.J., 1960. Foreword by Israel Halperin. MR 0120174
E. Tamás Schmidt, Zur Charakterisierung der Kongruenzverbände der Verbände, Mat. Časopis Sloven. Akad. Vied 18 (1968), 3–20 (German, with English and Russian summaries). MR 241335
E. Thomas Schmidt, The ideal lattice of a distributive lattice with $0$ is the congruence lattice of a lattice, Acta Sci. Math. (Szeged) 43 (1981), no. 1-2, 153–168. MR 621367
E. Tamás Schmidt, A survey on congruence lattice representations, Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], vol. 42, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1982. With German, French and Russian summaries. MR 668704
Michael Tischendorf, On the representation of distributive semilattices, Algebra Universalis 31 (1994), no. 3, 446–455. MR 1265355, DOI 10.1007/BF01221798
F. Wehrung, Non-measurability properties of interpolation vector spaces, Israel J. Math., to appear.