Lattice-Ordered Groups

1. Front Matter

   Pages i-ix

   PDF

2. Fundamentals

   • Marlow Anderson, Todd Feil

   Pages 1-14

3. Bernau’s Representation for Archimedean ℓ-groups

   • Marlow Anderson, Todd Feil

   Pages 15-20

4. The Conrad-Harvey-Holland Representation

   • Marlow Anderson, Todd Feil

   Pages 21-25

5. Representable and Normal-valued ℓ-groups

   • Marlow Anderson, Todd Feil

   Pages 26-31

6. Holland’s Embedding Theorem

   • Marlow Anderson, Todd Feil

   Pages 32-37

7. Free ℓ-groups

   • Marlow Anderson, Todd Feil

   Pages 38-46

8. Varieties of ℓ-groups

   • Marlow Anderson, Todd Feil

   Pages 47-63

9. Completions of Representable and Archimedean ℓ-groups

   • Marlow Anderson, Todd Feil

   Pages 64-76

10. The Lateral Completion

    • Marlow Anderson, Todd Feil

    Pages 77-85

11. Finite-valued and Special-valued ℓ-groups

    • Marlow Anderson, Todd Feil

    Pages 86-101

12. Groups of Divisibility

    • Marlow Anderson, Todd Feil

    Pages 102-108

13. Back Matter

    Pages 109-190

    PDF

The study of groups equipped with a compatible lattice order ("lattice-ordered groups" or "I!-groups") has arisen in a number of different contexts. Examples of this include the study of ideals and divisibility, dating back to the work of Dedekind and continued by Krull; the pioneering work of Hahn on totally ordered abelian groups; and the work of Kantorovich and other analysts on partially ordered function spaces. After the Second World War, the theory of lattice-ordered groups became a subject of study in its own right, following the publication of fundamental papers by Birkhoff, Nakano and Lorenzen. The theory blossomed under the leadership of Paul Conrad, whose important papers in the 1960s provided the tools for describing the structure for many classes of I!-groups in terms of their convex I!-subgroups. A particularly significant success of this approach was the generalization of Hahn's embedding theorem to the case of abelian lattice-ordered groups, work done with his students John Harvey and Charles Holland. The results of this period are summarized in Conrad's "blue notes" [C].

• Finite
• Morphism
• function
• theorem

• The Colorado College, Colorado Springs, USA

  Marlow Anderson

• Department of Mathematical Sciences, Denison University, Granville, USA

  Todd Feil

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