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algebra universalis

, Volume 3, Issue 1, pp 247–260 | Cite as

Archimedean lattices

  • J. Martinez
Article

Abstract

Anarchimedean lattice is a complete algebraic latticeL with the property that for each compact elementcL, the meet of all the maximal elements in the interval [0,c] is 0.L ishyper-archimedean if it is archimedean and for eachxL, [x, 1] is archimedean. The structure of these lattices is analysed from the point of view of their meet-irreducible elements. If the lattices are also Brouwer, then the existence of complements for the compact elements characterizes a particular class of hyper-archimedean lattices.

The lattice ofl-ideals of an archimedean lattice ordered group is archimedean, and that of a hyper-archimedean lattice ordered group is hyper-archimedean. In the hyper-archimedean case those arising as lattices ofl-ideals are fully characterized.

Keywords

Boolean Algebra Complete Lattice Algebra UNIV Universal Algebra Prime Element 
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.

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References

  1. [1]
    A. Bigard,Groupes archimédiens et hyper-archimédiens, No. 2 (1967–68).Google Scholar
  2. [2]
    G. Birkhoff,Lattice theory, Amer. Math. Soc. Coll. Publ.,XXV (1967).Google Scholar
  3. [3]
    G. Birkhoff and O. Frink,Representations of lattices by sets, Trans. Amer. Math. Soc.64 (1948), 299–316.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    R. Bleier and P. Conrad,The lattice of closed ideals and a *-extensions of an abelian l-group, preprint.Google Scholar
  5. [5]
    P. M. Cohn,Universal algebra, Harper and Row (1965).Google Scholar
  6. [6]
    P. Conrad,Lattice oddered groups, Tulane University (1970).Google Scholar
  7. [7]
    P. Conrad,Epi-archimedean lattice ordered groups, preprint.Google Scholar
  8. [8]
    O. Frink,Pseudo-complements in semi-lattices, Duke Math. Jour.29 (1962), 505–514.MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    G. Grätzer,Universal algebra, Van Nostrand (1968).Google Scholar
  10. [10]
    G. Grätzer and E. T. Schmidt,Characterizations of congruence lattices of abstract algebras, Acta Sci. Math.24 (1963), 34–59.MATHGoogle Scholar
  11. [11]
    J. P. Jans,Rings and homology, Holt, Rinehart and Winston (1964).Google Scholar
  12. [12]
    L. Nachbin,On a characterization of the lattice of all ideals of a Boolean ring, Fund. Math.36 (1949), 137–142.MATHMathSciNetGoogle Scholar
  13. [13]
    J. Varlet,Contribution à l’étude des treillis pseudo-complementés et des treillis de Stone, Mém. Soc. Roy. Sci. Liège8, (1963), 1–71.MathSciNetGoogle Scholar

Copyright information

© Birkhäuser-Verlag 1973

Authors and Affiliations

  • J. Martinez
    • 1
  1. 1.University of FloridaGainesvilleUSA

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