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Lattices

In: The Concise Handbook of Algebra

Author

Listed:
  • George Grätzer
  • E. Tamás Schmidt
  • Ralph Freese
  • Viacheslav N. Saliĭ
  • Gudrun Kalmbach
  • Carlton J. Maxson
  • Günter F. Pilz
  • Viacheslav N. Saliĭ
  • Rudolf Wille

Abstract

Let Con L denote, up to isomorphism, the class of congruence lattices of lattices and let DA denote the class of all distributive algebraic lattices. For every lattice L, it it clear that the congruence lattice Con L is algebraic. By a 1942 result of N. Funayama and T. Nakayama, Con L is also distributive, so Con L ⊆ DA. Is the converse true: Is every distributive algebraic lattice isomorphic to the congruence lattice of a suitable lattice? This is one of the most famous open questions of lattice theory. We shall briefly review this topic here, together with its related results; for a more complete overview (up to 1998), see Appendix C in (Grätzer 1998); we shall only reference later papers here.

Suggested Citation

  • George Grätzer & E. Tamás Schmidt & Ralph Freese & Viacheslav N. Saliĭ & Gudrun Kalmbach & Carlton J. Maxson & Günter F. Pilz & Viacheslav N. Saliĭ & Rudolf Wille, 2002. "Lattices," Springer Books, in: The Concise Handbook of Algebra, chapter 0, pages 417-450, Springer.
    • Handle: RePEc:spr:sprchp:978-94-017-3267-3_6
    DOI: 10.1007/978-94-017-3267-3_6
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