Surveying Lattice-Ordered Fields

  • R. H. Redfield
Part of the Developments in Mathematics book series (DEVM, volume 7)

Abstract

The object of what follows is to give a brief overview of the theory of lattice-ordered fields. While I have included no proofs, I have tried to give ample references for anyone interested in seeing the details. Section 1 briefly sketches the history behind the subject and section 2 recalls some basic definitions. The remainder reviews what is presently known: section 3 describes methods of constructing lattice-ordered fields; section 4 concerns the maximal totally ordered subfield; section 5 considers lattice-ordered fields as vector lattices; section 6 describes representations of lattice-ordered fields by means of power series fields; section 7 discusses the number of different compatible lattice orderings; section 8 deals with extensions to total orders; section 9 investigates the lattice-ordering of simple algebraic extensions of totally ordered fields; and section 10 lists some open questions. Of course not all results are mentioned below. I have chosen those that seem to me to be the most fundamental and the most interesting.

I have cited specific results in one of two ways: if an author has given the n th result in the m th section the number m.n, then I have used that number; otherwise, I have used the number of the page on which the result occurs.

Keywords

Vector Lattice Total Order Convex Subgroup Positive Multiplicative Inverse Supporting Collection 
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]
    M. Anderson and T. Feil, Lattice-Ordered Groups: An Introduction. (1987) D. Reidel, Dordrecht, ISBN 90–277-2643–4.Google Scholar
  2. [2]
    E. Artin, Über die Zerlegung definiter Funktionen in Quadrate. Abh. Math. Sem. Hamb. Univ. 5 (1926), 100–115.MATHCrossRefGoogle Scholar
  3. [3]
    E. Artin and O. Schreier, Algebraische Konstruktion reëler Körper. Abh. Math. Sem. Hamb. Univ. 5 (1926), 85–99.MATHCrossRefGoogle Scholar
  4. [4]
    A. Bigard, K. Keimei and S. Wolfenstein, Groupes et Anneaux Réticulés. Lecture Notes in Mathematics 608 (1977), Springer-Verlag, Berlin, ISBN 3–540-08436–3.MATHGoogle Scholar
  5. [5]
    G. Birkhoff, Lattice-Ordered Groups. Annals of Math. 43 (1942), 298–331.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    G. Birkhoff, Lattice Theory. 3rd ed. (AMS Coll. Pub. 25) (1973), Amer. Math. Soc, Providence.Google Scholar
  7. [7]
    G. Birkhoff and R. S. Pierce, Lattice-ordered rings. An. Acad. Brasil Ci. 28 (1956), 41–69.MathSciNetMATHGoogle Scholar
  8. [8]
    N. Bourbaki, Eléments d’Histoire des Mathématiques. (1969) Hermann, Paris.MATHGoogle Scholar
  9. [9]
    P. Conrad, Generalized semigroup rings. J. Indian Math. Soc. 21 (1957), 73–95.MathSciNetGoogle Scholar
  10. [10]
    P. Conrad, Lattice Ordered Groups. Tulane University (1970), New Orleans.MATHGoogle Scholar
  11. [11]
    P. Conrad and J. Dauns, An embedding theorem for lattice-ordered fields. Pacific J. Math. 3 (1969), 385–398.MathSciNetCrossRefGoogle Scholar
  12. [12]
    P. Conrad, J. Harvey and W. C. Holland, The Hahn embedding theorem for lattice-ordered groups. Trans. Amer. Math. Soc. 108 (1963), 143–169.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    M. R. Darnel, The Theory of Lattice-Ordered Groups. (1995) Marcel Dekker, New York, ISBN 0–8247-9326–9.Google Scholar
  14. [14] R. Dedekind, Über Zerlegungen von Zahlen durch ihre grossen ten Gemeinsamen Teiler. Gesammelte Math. Werke, t. II (1931), 103–147; (originally appeared in Festschrift der Techn. Hochsch. zu Braunschweig bei Gelegenheit der 69 Versammlung Deutscher Naturforscher und Ärtze (1897), 1–40.) [15] R. DeMarr, Partially ordered fields. Amer. Math. Monthly 74 (1967), 418–420.Google Scholar
  15. [16]
    R. DeMarr, A. Steiger, On elements with negative squares. Proc. Amer. Math. Soc. 31 (1972), 57–60.MathSciNetMATHCrossRefGoogle Scholar
  16. [17]
    D. W. Dubois, On partly ordered fields. Proc. Amer. Math. Soc. 7 (1956), 918–930.MathSciNetMATHCrossRefGoogle Scholar
  17. [18]
    H. Freudenthal, Teilweise geordnete moduln. Proc. Ned. Akad. Wet. 39 (1936), 641–651.Google Scholar
  18. [19]
    L. Fuchs, Partially Ordered Algebraic Systems. (1963) Pergamon Press, Oxford.MATHGoogle Scholar
  19. [20]
    A. M. W. Glass, Partially Ordered Groups. (1999) World Scientific, Singapore, ISBN 981–02-3493–7.MATHCrossRefGoogle Scholar
  20. [21]
    H. Hahn, Über die nichtarchimedean Grössensysteme. Sitzungber. Kaiserlichen Akad. Wiss. Vienna Math. Nat. Klasse Abt. IIa 116 (1907), 601–653.MATHGoogle Scholar
  21. [22]
    M. Henriksen, On the difficulties in embedding lattice-ordered integral domains in lattice-ordered fields. General Topology and its Relations to Modern Analysis and Algebra III (Proc. Third Prague Topological Sympos., 1971) (1972) Academia, Prague, 183–185.Google Scholar
  22. [23]
    S. Kakutani, Concrete representation of abstract (L)-spaces and the mean er-godic theorem. Annals of Math. 42 (1941), 523–537.MathSciNetCrossRefGoogle Scholar
  23. [24]
    S. Kakutani, Concrete representation of abstract (M)-spaces (a characterization of the space of continuous functions). Annals of Math. 42 (1941), 994–1024.MathSciNetMATHCrossRefGoogle Scholar
  24. [25]
    L. V. Kantorovich, Partially ordered linear spaces. (Russian) Mat. Sbornik 2 (44) (1937), 121–168.MATHGoogle Scholar
  25. [26]
    S. MacLane, Categories for the Working Mathematician. (1971) Springer-Verlag, New York, ISBN 0–387-90035–7.Google Scholar
  26. [27]
    B. Neumann, On ordered division rings. Trans. Amer. Math. Soc. 66 (1949), 202–252.MathSciNetMATHCrossRefGoogle Scholar
  27. [28]
    S. Prieß-Crampe, Angeordnete Strukturen: Gruppen, Körper, Projektive Ebenen. Ergeb. der Math. u. i. Grenzgeb. 98 (1983) Springer-Verlag, Berlin.MATHGoogle Scholar
  28. [29]
    F. J. Raynor, An algebraically closed field. Glasgow Math. J. 9 (1968) 146–151.MathSciNetCrossRefGoogle Scholar
  29. [30]
    R. H. Redfield, Algebraic extensions and lattice-ordered fields. Analysis Paper No. 15 (1975) Monash Univ., Clayton, Victoria, Australia.Google Scholar
  30. [31]
    R. H. Redfield, Embeddings into power series rings. Manuscripta Math. 56 (1986), 247–268.MathSciNetMATHCrossRefGoogle Scholar
  31. [32]
    R. H. Redfield, Constructing lattice-ordered fields and division rings. Bull. Austr. Math. Soc. 40 (1989), 365–369.MathSciNetMATHCrossRefGoogle Scholar
  32. [33]
    R. H. Redfield, Lattice-ordered fields as convolution algebras. J. Algebra 153 (1992) 319–356.MathSciNetMATHCrossRefGoogle Scholar
  33. [34]
    R. H. Redfield, Lattice-ordered power series fields. J. Austr. Math. Soc. (Series A) 52 (1992), 229–321.MathSciNetCrossRefGoogle Scholar
  34. [35]
    R. H. Redfield, Internal characterizations of lattice-ordered power series fields. Per. Math. Hungarica 32 (1996), 85–101.MathSciNetMATHCrossRefGoogle Scholar
  35. [36]
    R. H. Redfield, Abstract Algebra: A Concrete Introduction. (2001) Addison Wesley Longman, Boston, ISBN 0–201-43721-X.Google Scholar
  36. [37]
    R. H. Redfield, Subfields of lattice-ordered fields that mimic maximal totally ordered subfields. Czech. Math. J. 51 (126) (2001), 143–161.MathSciNetMATHCrossRefGoogle Scholar
  37. [38]
    R. H. Redfield, Unexpected lattice-ordered quotient structures. Ordered Algebraic Structures: Nanjing (2001); W. C. Holland, ed.; Gordon and Breach, The Netherlands, ISBN 90–5699-325–9, 111–132.Google Scholar
  38. [39]
    R. H. Redfield, Letter to Stuart Steinberg. (2001).Google Scholar
  39. [40]
    P. Ribenboim, Rings of generalized power series: nilpotent elements. Abh. Math. Sem. Univ. Hamburg (Series A) 61 (1991), 15–33.MathSciNetMATHCrossRefGoogle Scholar
  40. [41]
    P. Ribenboim, Noetherian rings of generalized power series. J. Pure Appl. Algebra 79 (1992), 293–312.MathSciNetMATHCrossRefGoogle Scholar
  41. [42]
    P. Ribenboim, Rings of generalized power series, II: units and zero-divisors. J. Algebra 168 (1994), 71–89.MathSciNetMATHCrossRefGoogle Scholar
  42. [43]
    F. Riesz, Sur quelques notions fondamentales dans la théorie générale des opérations linéaires. Annals of Math 41 (1940), 174–206.MathSciNetCrossRefGoogle Scholar
  43. [44]
    N. Schwartz, Verbandsgeordnete Körper. (1978) Dissertation, Universität München.Google Scholar
  44. [45]
    N. Schwartz, Archimedean lattice-ordered fields that are algebraic over their o-subfields. Pacific J. Math. 89 (1980), 189–198.MathSciNetCrossRefGoogle Scholar
  45. [46]
    N. Schwartz, Lattice-ordered fields. Order 3 (1986), 179–194.MathSciNetMATHCrossRefGoogle Scholar
  46. [47]
    J.-P. Serre, Extensions des corps ordonnés. C. R. Acad. Sci. Paris 229 (1949), 576–577.MathSciNetMATHGoogle Scholar
  47. [48]
    S. Steinberg, An embedding theorem for commutative lattice-ordered domains. Proc. Amer. Math. Soc. 31 (1972), 409–416.MathSciNetMATHCrossRefGoogle Scholar
  48. [49]
    S. Steinberg, Personal Communication. (1990).Google Scholar
  49. [50]
    R. R. Wilson, Lattice orderings on fields and certain rings. Symposia Mathematica (Convegno sulle Misure su Gruppi e su Spazi Vettoriali, Convegno sui Gruppi e Anelli Ordinati, INDAM, Rome) 21 (1975), 357–364.Google Scholar
  50. [51]
    R. R. Wilson, Lattice orderings on the real field. Pacific J. Math. 63 (1976), 571–577.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2002

Authors and Affiliations

  • R. H. Redfield
    • 1
  1. 1.Department of MathematicsHamilton CollegeClintonUSA

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