Ideals in Birkhoff Lattices

  • R. P. Dilworth
Part of the Contemporary Mathematicians book series (CM)


In previous papers by the author (Dilworth [1, 2])(2) methods were developed for studying the arithmetical properties of Birkhoff lattices, that is, the properties of irreducibles and decompositions into irreducibles. These methods, however, required the assumption of both the ascending and descending chain conditions. In this paper we give a new technique which is applicable in general and which under the assumption of merely the ascending chain condition gives results quite as good as those of the previous work. Now the descending chain condition is equivalent to the requirement that every ideal(3) be principal. Hence if the descending chain condition does not hold we find it convenient to relate the arithmetical properties of the lattice to the structure of its lattice of ideals. Furthermore since the Birkhoff condition itself may lose much of its force if the descending chain condition does not hold, a lattice is defined to be a Birkhoff lattice if every element satisfies the Birkhoff condition(4) in the lattice of ideals. Hence if the descending chain condition holds, this definition reduces to that used in the previous papers. In the lattice of ideals, the existence of sufficient covering ideals to make the Birkhoff conditions effective can be proved.


Boolean Algebra Point Ideal Principal Ideal Chain Condition Characteristic Ideal 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    G. Birkhoff Rings of sets, Duke Mathematical Journal, vol. 3 (1937), pp. 443–454.CrossRefGoogle Scholar
  2. 2.
    G. Birkhoff On combination of subalgebras, Proceedings of the Cambridge Philosophical Society, vol. 29 (1933), pp. 441–464.CrossRefGoogle Scholar
  3. 1.
    R.P. Dilworth Lattices with unique irreducible decompositions, Annals of Mathematics.Google Scholar
  4. 2.
    R.P. Dilworth The arithmetical theory of Birkhoff lattices, submitted to Duke Mathematical Journal.Google Scholar
  5. 1.
    S. MacLane A lattice formulation for transcendence degrees and p-bases, Duke Mathematical Journal, vol. 24 (1938), pp. 455–468.CrossRefGoogle Scholar
  6. 1.
    M.H. Stone The theory of representations of Boolean algebra, these Transactions, vol. 40 (1936), pp. 37–111.Google Scholar
  7. 1.
    H. Wallman Lattices and topological spaces, Annals of Mathematics, (2), vol. 39 (1938), pp. 112–126.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1990

Authors and Affiliations

  • R. P. Dilworth
    • 1
  1. 1.Yale UniversityNew HavenUSA

Personalised recommendations