Remote Access Proceedings of the American Mathematical Society
Green Open Access

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

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 G. M. Bergman, Von Neumann regular rings with tailor-made ideal lattices, unpublished note (1986).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. G. Birkhoff, Lattice theory, Corrected reprint of the 1967 third edition. American Mathematical Society Colloquium Publications, 25. American Mathematical Society, Providence, R.I., 1979. vi+418 pp. MR 82a:06001
  • 4. H. Dobbertin, Refinement monoids, Vaught Monoids, and Boolean Algebras, Math. Ann. 265 (1983), pp. 475-487. MR 85e:06016
  • 5. K. R. Goodearl, von Neumann regular rings, Second edition. Robert E. Krieger Publishing Co., Inc., Malabar, FL, 1991. xviii+412 pp. MR 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. G. Grätzer, Lattice Theory: first concepts and distributive lattices, W. H. Freeman, San Francisco, Cal., 1971. MR 48:184
  • 8. G. Grätzer, General Lattice Theory, Pure and Applied Mathematics 75, Academic Press, Inc.(Harcourt Brace Jovanovich, Publishers), New York-London; Lehrbücher und Monographien aus dem Gebiete der Exakten Wissenschaften, Mathematische Reihe, Band 52. Birkhäuser Verlag, Basel-Stuttgart; Akademie Verlag, Berlin, 1978. xiii+381 pp. MR 80c:06001b; MR 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. J. von Neumann, Continuous geometry, Foreword by Israel Halperin. Princeton Mathematical Series, No. 25 Princeton University Press, Princeton, N. J. 1960 xi+299 pp. MR 22:10931
  • 11. E. T. Schmidt, Zur Charakterisierung der Kongruenzverbände der Verbände, Mat. \v{C}asopis Sloven. Akad. Vied. 18 (1) (1968), pp. 3-20. MR 39:2675
  • 12. E. T. Schmidt, The ideal lattice of a distributive lattice with $0$ is the congruence lattice of a lattice, Acta Sci. Math. (Szeged) 43 (1981), pp. 153-168. MR 82g:06015
  • 13. E. T. Schmidt, A survey on congruence lattice representations, Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], 42. BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1982. 115 pp. MR 84c:06012
  • 14. M. Tischendorf, On the representation of distributive semilattices, Algebra Universalis 31 (1994), pp. 446-455. MR 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

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

American Mathematical Society