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The Theory of Lattice-Ordered Groups

  • V. M. Kopytov
  • N. Ya. Medvedev

Part of the Mathematics and Its Applications book series (MAIA, volume 307)

Table of contents

  1. Front Matter
    Pages i-xvi
  2. V. M. Kopytov, N. Ya. Medvedev
    Pages 1-9
  3. V. M. Kopytov, N. Ya. Medvedev
    Pages 11-29
  4. V. M. Kopytov, N. Ya. Medvedev
    Pages 31-50
  5. V. M. Kopytov, N. Ya. Medvedev
    Pages 51-90
  6. V. M. Kopytov, N. Ya. Medvedev
    Pages 91-110
  7. V. M. Kopytov, N. Ya. Medvedev
    Pages 111-131
  8. V. M. Kopytov, N. Ya. Medvedev
    Pages 133-160
  9. V. M. Kopytov, N. Ya. Medvedev
    Pages 161-185
  10. V. M. Kopytov, N. Ya. Medvedev
    Pages 187-236
  11. V. M. Kopytov, N. Ya. Medvedev
    Pages 237-254
  12. V. M. Kopytov, N. Ya. Medvedev
    Pages 255-281
  13. V. M. Kopytov, N. Ya. Medvedev
    Pages 283-334
  14. V. M. Kopytov, N. Ya. Medvedev
    Pages 335-343
  15. V. M. Kopytov, N. Ya. Medvedev
    Pages 345-377
  16. Back Matter
    Pages 379-400

About this book

Introduction

A partially ordered group is an algebraic object having the structure of a group and the structure of a partially ordered set which are connected in some natural way. These connections were established in the period between the end of 19th and beginning of 20th century. It was realized that ordered algebraic systems occur in various branches of mathemat­ ics bound up with its fundamentals. For example, the classification of infinitesimals resulted in discovery of non-archimedean ordered al­ gebraic systems, the formalization of the notion of real number led to the definition of ordered groups and ordered fields, the construc­ tion of non-archimedean geometries brought about the investigation of non-archimedean ordered groups and fields. The theory of partially ordered groups was developed by: R. Dedekind, a. Holder, D. Gilbert, B. Neumann, A. I. Mal'cev, P. Hall, G. Birkhoff. These connections between partial order and group operations allow us to investigate the properties of partially ordered groups. For exam­ ple, partially ordered groups with interpolation property were intro­ duced in F. Riesz's fundamental paper [1] as a key to his investigations of partially ordered real vector spaces, and the study of ordered vector spaces with interpolation properties were continued by many functional analysts since. The deepest and most developed part of the theory of partially ordered groups is the theory of lattice-ordered groups. In the 40s, following the publications of the works by G. Birkhoff, H. Nakano and P.

Keywords

Group theory Lattice algebra semigroup

Authors and affiliations

  • V. M. Kopytov
    • 1
  • N. Ya. Medvedev
    • 2
  1. 1.Institute of Mathematics, Siberian BranchRussian Academy of SciencesNovosibirskRussia
  2. 2.Altai State UniversityBarnaulRussia

Bibliographic information

  • DOI https://doi.org/10.1007/978-94-015-8304-6
  • Copyright Information Springer Science+Business Media B.V. 1994
  • Publisher Name Springer, Dordrecht
  • eBook Packages Springer Book Archive
  • Print ISBN 978-90-481-4474-7
  • Online ISBN 978-94-015-8304-6
  • Buy this book on publisher's site
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