Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


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
Full-text PDF Free Access

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 $A$ 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.

References [Enhancements On Off] (What's this?)

  • 1. 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.
  • 2. G. M. Bergman, Von Neumann regular rings with tailor-made ideal lattices, unpublished note (1986).
  • 3. Garrett Birkhoff, Lattice theory, 3rd ed., American Mathematical Society Colloquium Publications, vol. 25, American Mathematical Society, Providence, R.I., 1979. MR 598630 (82a:06001)
  • 4. Hans Dobbertin, Refinement monoids, Vaught monoids, and Boolean algebras, Math. Ann. 265 (1983), no. 4, 473–487. MR 721882 (85e:06016),
  • 5. K. R. Goodearl, von Neumann regular rings, 2nd ed., Robert E. Krieger Publishing Co., Inc., Malabar, FL, 1991. MR 1150975 (93m:16006)
  • 6. 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. CMP 96:09
  • 7. George Grätzer, Lattice theory. First concepts and distributive lattices, W. H. Freeman and Co., San Francisco, Calif., 1971. MR 0321817 (48 #184)
  • 8. George Grätzer, General lattice theory, Pure and Applied Mathematics, vol. 75, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 509213 (80c:06001b)
    George Grätzer, General lattice theory, Birkhäuser Verlag, Basel-Stuttgart, 1978. Lehrbücher und Monographien aus dem Gebiete der Exakten Wissenschaften, Mathematische Reihe, Band 52. MR 504338 (80c:06001a)
  • 9. G. Grätzer and E.T. Schmidt, Congruence lattices of lattices, Appendix C in G. Grätzer, General Lattice Theory, Second Edition, to appear.
  • 10. John von Neumann, Continuous geometry, Foreword by Israel Halperin. Princeton Mathematical Series, No. 25, Princeton University Press, Princeton, N.J., 1960. MR 0120174 (22 #10931)
  • 11. E. Tamás Schmidt, Zur Charakterisierung der Kongruenzverbände der Verbände, Mat. Časopis Sloven. Akad. Vied 18 (1968), 3–20 (German, with Loose English and Russian summaries). MR 0241335 (39 #2675)
  • 12. 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 (82g:06015)
  • 13. 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 (84c:06012)
  • 14. Michael Tischendorf, On the representation of distributive semilattices, Algebra Universalis 31 (1994), no. 3, 446–455. MR 1265355 (95g:06010),
  • 15. F. Wehrung, Non-measurability properties of interpolation vector spaces, Israel J. Math., to appear.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 06A12, 06B10, 16E50

Retrieve articles in all journals with MSC (1991): 06A12, 06B10, 16E50

Additional Information

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

PII: S 0002-9939(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