Lattice-Ordered Groups: An Introduction

✍ Scribed by Marlow Anderson, Todd Feil (auth.)

• Publisher: Springer Netherlands
• Year: 1988
• Tongue: English
• Leaves: 196
• Series: Reidel Texts in the Mathematical Sciences 4
• Edition: 1
• Category: Library

✦ Synopsis

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].

✦ Table of Contents

Front Matter....Pages i-ix
Fundamentals....Pages 1-14
Bernau’s Representation for Archimedean ℓ-groups....Pages 15-20
The Conrad-Harvey-Holland Representation....Pages 21-25
Representable and Normal-valued ℓ-groups....Pages 26-31
Holland’s Embedding Theorem....Pages 32-37
Free ℓ-groups....Pages 38-46
Varieties of ℓ-groups....Pages 47-63
Completions of Representable and Archimedean ℓ-groups....Pages 64-76
The Lateral Completion....Pages 77-85
Finite-valued and Special-valued ℓ-groups....Pages 86-101
Groups of Divisibility....Pages 102-108
Back Matter....Pages 109-190

✦ Subjects

Discrete Mathematics in Computer Science