Hull Classses of Archimedean Lattice-Ordered Groups with Unit: A Survey

  • Jorge Martínez
Part of the Developments in Mathematics book series (DEVM, volume 7)


This is a survey of the literature on hull classes of archimedean lattice-ordered groups with a designated unit. There has been a substantial amount of activity in this specialty in the last decade, and the goal here is to put the subj ect in some perspective, with an account of some of the history of accomplishments, as well as of the most recent progress.


Compact Space Riesz Space Semiprime Ring Essential Extension Tychonoff Space 


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  1. [An65]
    F. W. Anderson, Lattice-ordered rings of quotients. Canad. Jour. Math. 17 (1965), 434–448.MATHCrossRefGoogle Scholar
  2. [AF88]
    M. Anderson & T. Feil, Lattice-Ordered Groups, an Introduction. (1988) Reidel Texts in the Math. Sci.; Kluwer, Dordrecht.MATHCrossRefGoogle Scholar
  3. [Ar71]
    E. R. Aron, Embedding lattice-ordered algebras in uniformly closed algebras. (1971) Thesis, University of Rochester.Google Scholar
  4. [ArH81]
    E. R. Aron & A. W. Hager, Convex vector lattices and ℓ-algebras. Topology and its Appl. 12 (1981), 1–10.MathSciNetMATHCrossRefGoogle Scholar
  5. [BH89]
    R. N. Ball & A. W. Hager, Epimorphisms in archimedean lattice-ordered groups and vector lattices. In Lattice-Ordered Groups, Advances and Techniques, (A. M. W. Glass & W. C. Holland, Eds.); Math. and its Appl. (1989); Kluwer Acad. Publ., Dordrecht.Google Scholar
  6. [BH90a]
    R. N. Ball & A. W. Hager, Epimorphisms in archimedean ℓ-groups and vector lattices with weak unit (and Baire functions). J. Austral. Math. Soc. (Ser. A) 48 (1990), 351–368.MathSciNetCrossRefGoogle Scholar
  7. [BH90b]
    R. N. Ball & A. W. Hager, Epicomplete archimedean ℓ-groups and vector lattices. Trans. AMS 322 (No. 2) (1990), 459–478.MathSciNetMATHGoogle Scholar
  8. [BH91]
    R. N. Ball & A. W. Hager, On the localic Yosida representation of an archimedean lattice-ordered group with weak unit. J. of Pure & Appl. Alg. 70 (1991), 17–43.MathSciNetMATHCrossRefGoogle Scholar
  9. [BH93]
    R. N. Ball & A. W. Hager, Algebraic extensions of an archimedean latticeordered group, I. J. of Pure & Appl. Algebra 85 (1993), 1–20.MathSciNetMATHCrossRefGoogle Scholar
  10. [BH99a]
    R. N. Ball & A. W. Hager, Algebraic extensions of an archimedean latticeordered group, II. J. of Pure & Appl. Algebra 138 (1999), 197–204.MathSciNetMATHCrossRefGoogle Scholar
  11. [BH99b]
    R. N. Ball & A. W. Hager, The relative uniform density of the continuous functions in the Baire functions, and of a divisible archimedean ℓ-group in any epicompletion. Topology and its Appl. 97 (1999), 109–126.MathSciNetMATHCrossRefGoogle Scholar
  12. [Ba65]
    B. Banaschewski, Maximal rings of quotients of semi-simple commutative rings. Archiv. Math. XVI (1965), 414–420.MathSciNetCrossRefGoogle Scholar
  13. [Be75a]
    S. J. Bernau, The lateral completion of a lattice ordered group. J. Austral. Math. Soc. 19 (1975), 263–289.MathSciNetMATHCrossRefGoogle Scholar
  14. [Be75b]
    S. J. Bernau, Lateral and Dedekind completion of archimedean lattice groups. J. London Math. Soc. 12 (1975/76), 320–322.MathSciNetCrossRefGoogle Scholar
  15. [BKW77]
    A. Bigard, K. Keimel & S. Wolfenstein, Groupes et Anneaux Réticulés. Lecture Notes in Math 608, Springer Verlag (1977); Berlin-Heidelberg-New York.Google Scholar
  16. [Bi67]
    G. Birkhoff, Lattice Theory (3rd Ed.) AMS Colloq. Publ. XXV (1967), Providence, RI.MATHGoogle Scholar
  17. [B174]
    R. D. Bleier, The SP-hull of a lattice-ordered group. Canad. Jour. Math. XXVI, No. 4 (1974), 866–878.MathSciNetCrossRefGoogle Scholar
  18. [Ch71]
    D. Chambless, The Representation and Structure of Lattice-Ordered Groups an f-Rings. Tulane University Dissertation (1971), New Orleans.Google Scholar
  19. [C69]
    P. F. Conrad, The lateral completion of a lattice-ordered group. Proc. London Math. Soc. 3rd Series XIX (July 1969), 444–480.MathSciNetCrossRefGoogle Scholar
  20. [C71]
    P. F. Conrad, The essential closure of an archimedean lattice-ordered group. Duke Math. Jour. 38 (1971), 151–160.MathSciNetMATHCrossRefGoogle Scholar
  21. [C73]
    P. F. Conrad, The hulls of representable ℓ-groups and f-rings. J. Austral. Math. Soc. 26 (1973), 385–415.MathSciNetCrossRefGoogle Scholar
  22. [CMc69]
    P. F. Conrad & D. McAlister, The completion of a lattice-ordered group. J. Austral. Math. Soc. 9 (1969), 182–208.MathSciNetMATHCrossRefGoogle Scholar
  23. [D95]
    M. R. Darnel, The Theory of Lattice-Ordered Groups. Pure & Appl. Math. 187, Marcel Dekker (1995); Basel-Hong Kong-New York.Google Scholar
  24. [DHH80]
    F. Dashiell, A. W. Hager & M. Henriksen, Order-Cauchy completions of rings and vector lattices of continuous functions. Canad. Jour. Math. 32 (1980), 657–685.MathSciNetMATHCrossRefGoogle Scholar
  25. [Ev44]
    C. J. Everett, Sequence completions of lattice modules. Duke Math. J. 11 (1944), 109–119.MathSciNetMATHCrossRefGoogle Scholar
  26. [FGL65]
    N. Fine, L. Gillman & J. Lambek, Rings of Quotients of Rings of Functions. (1965) McGill University.Google Scholar
  27. [Fu63]
    L. Fuchs, Partially Ordered Algebraic Systems. (1963) Pergamon Press, Oxford-New York-London-Paris.MATHGoogle Scholar
  28. [GJ76]
    L. Gillman & M. Jerison, Rings of Continuous Functions. Grad. Texts in Math. 43, Springer Verlag (1976); Berlin-Heidelberg-New York.Google Scholar
  29. [G158]
    A. M. Gleason, Projective topological spaces. Illinois J. Math. 7 (1958), 482–489.MathSciNetGoogle Scholar
  30. [Gv29]
    V. Glivenko, Bull. Acad. Sci. Belg. 15 (1929), 183–88.MATHGoogle Scholar
  31. [H85]
    A. W. Hager, Algebraic closures of ℓ-groups of continuous functions. In Rings of Continuous Functions; C. Aull, Ed.; Lecture Notes in Pure & Appl. Math. 95 (1985), Marcel Dekker, New York, 165–194.Google Scholar
  32. [H89]
    A. W. Hager, Minimal covers of topological spaces. In Papers on General Topology and Related Category Theory and Topological Algebra; Annals of the N. Y. Acad. Sci. 552 March 15, 1989, 44–59.MathSciNetGoogle Scholar
  33. [HM93]
    A. W. Hager & J. Martínez, Functorial rings of quotients, I. In Proc. Conf. Ord. Alg. Struc. ;(W. C. Holland & J. MartInez, Eds.); Gainesville, 1991; (1993) Kluwer Acad. Publ., Dordrecht, 133–157.Google Scholar
  34. [HM94a]
    A. W. Hager & J. Martínez, Functorial rings of quotients, II. Forum Math. 6 (1994), 597–616.MathSciNetMATHCrossRefGoogle Scholar
  35. [HM94b]
    A. W. Hager & J. Martinez, Maximum monoreflections. Appl. Categ. Struc. 2 (1994), 315–329.MathSciNetMATHCrossRefGoogle Scholar
  36. [HM96]
    A. W. Hager & J. Martinez, a-Projectable and laterally a complete Archimedean lattice-ordered groups. In Proc. Conf. in Memory of T. Retta; S. Bernahu, ed.;(1995)Google Scholar
  37. [HM96]
    A. W. Hager & J. Martinez, Temple U. Ethiopian J. Sci. (1996), 73–84.Google Scholar
  38. [HM97]
    A. W. Hager & J. Martínez, The laterallyσ-complete reflectionlection of an archimedean lattice-ordered group. In Ord. Alg. Struc. ; Cura cao (1995), (W. C. Holland & J. Martínez, Eds.); (1997) Kluwer Acad. Publ., Dordrecht, 217–236.Google Scholar
  39. [HM98a]
    A. W. Hager & J. Martínez, Pushout-invariant extensions and monoreflections. J. of Pure and Appl. Alg. 129 (1998), 263–295.MATHCrossRefGoogle Scholar
  40. [HM98b]
    A. W. Hager & J. Martínez, Singular archimedean lattice-ordered groups. Alg. Universalis 40 (1998), 119–147.MATHCrossRefGoogle Scholar
  41. [HM99a]
    A. W. Hager & J. Martínez, More on the laterally o-complete reflection of an archimedean lattice-ordered group. Order 15 (1999), 247–260.CrossRefGoogle Scholar
  42. [HM99b]
    A. W. Hager & J. Martínez, Hulls for various kinds of a-completeness in archimedean lattice-ordered groups. Order 16 (1999), 89–103.MathSciNetMATHCrossRefGoogle Scholar
  43. [HM01C]
    A. W. Hager & J. Martínez, Functorial approximation to the lateral completion in archimedean lattice-ordered groups with unit. Rend. Sem. Mat. Univ. Padova 105 (2001), 87–110.MathSciNetMATHGoogle Scholar
  44. [HMO1b]
    A. W. Hager & J. Martínez, Maximum monoreflectionslections and essential extensions. Appl. Categ. Struc. 9 (2001), 517–523.MATHCrossRefGoogle Scholar
  45. [HMO1c]
    A. W. Hager & J. Martinez, Functorial rings of quotients, III: the maximum in archimedean f-rings. To appear; J. of Pure & Appl. Alg.Google Scholar
  46. [HMO1d]
    A. W. Hager & J. Martínez, The ring of a-quotients. To appear, Algebra Universalis.Google Scholar
  47. [HM∞a]
    A. W. Hager & J. Martinez, On strong a-regularity. Work in progress.Google Scholar
  48. [HM∞b]
    A. W. Hager & J. Martínez, Polar functions, II: Completion classes of archimedean f-algebras vs. covers of compact spaces. Preprint.Google Scholar
  49. [HM∞c] A. W. Hager & J. Martinez, Polar functions, III: On irreducible maps vs essential extensions of archimedean ℓ-groups with unit. Work in progress.Google Scholar
  50. [HM∞d]
    A. W. Hager & J. Martínez, The projectable and regular hulls of a semiprime ring. Preprint.Google Scholar
  51. [HM∞e]
    A. W. Hager & J. Martinea, The regular reflection of a semiprime f-ring. Work in progress.Google Scholar
  52. [HR77]
    A. W. Hager & L. C. Robertson, Representing and ringifying a Riesz space. Symp. Math. 21 (1977), 411–431.MathSciNetGoogle Scholar
  53. [HR78]
    A. W. Hager & L. C. Robertson, Extremal units in an archimedean Riesz space. Rend. Sem. Mat. Univ. Padova 59 (1978), 97–115.MathSciNetMATHGoogle Scholar
  54. [HR79]
    A. W. Hager & L. C. Robertson, On the embedding into a ring of an archimedean ℓ-group. Canad. J. Math. 31 (1979), 1–8.MathSciNetMATHCrossRefGoogle Scholar
  55. [HIJ61]
    M. Henriksen, J. R. Isbell & D. G. Johnson, Residue class fields of latticeordered algebras. Fund. Math. 50 (1961), 107–117.MathSciNetMATHGoogle Scholar
  56. [HJ61]
    M. Henriksen & D. G. Johnson, On the structure of a class of lattice-ordered algebras. Fund. Math. 50 (1961), 73–94.MathSciNetMATHGoogle Scholar
  57. [HVW87]
    M. Henriksen, J. Vermeer & R. G. Woods, The quasi F-cover of Tychonoff spaces. Trans. AMS 303 (1987), 779–803.MathSciNetMATHGoogle Scholar
  58. [HVW89]
    M. Henriksen, J. Vermeer & R. G. Woods, Wallman covers of compact spaces. Diss. Math. CCLXXX (1989), Warsaw.Google Scholar
  59. [HS79]
    H. Herrlich & G. E. Strecker, Category Theory. Sigma Series in Pure Math. 1 (1979), Heldermann Verlag, Berlin.MATHGoogle Scholar
  60. [HP82]
    C. B. Huismans & B. de Pagter, Ideal theory in f-algebras. Trans. AMS 269 (No. 1) (January, 1982), 225–245.Google Scholar
  61. [HP83]
    C. B. Huijsmans & B. de Pagter, Maximal d-ideals in a Riesz space. Canad. Jour. Math. 35 (1983), 1010–1029.MATHCrossRefGoogle Scholar
  62. [L86]
    J. Lambek, Lectures on Rings and Modules. (3rd Ed.) (1986) Chelsea Publ. Co., New York.Google Scholar
  63. [LZ71]
    W. A. J. Luxemburg & A. C. Zaanen, Riesz Spaces, I. (1971) North Holland, Amsterdam.Google Scholar
  64. [MV90]
    J. J. Madden & J. Vermeer, Epicomplete archimedean ℓ-groups via a localic Yosida theorem. J. of Pure & Appl. Alg. 68 (1990), 243–252.MathSciNetMATHCrossRefGoogle Scholar
  65. [M95]
    J. Martinez, The maximal ring of quotients of an f-ring. Alg. Universalis 33 (1995), 335–369.Google Scholar
  66. [M02]
    J. Martfnez, Polar functions, I: The summand-inducing hull of an archimedean lattice-ordered group with unit. In these proceedings.Google Scholar
  67. [dP81]
    B. de Pagter, On z-ideals and d-ideals in Riesz spaces, III. Proc. Kon. Nederl. Akad. Wetensch., Series A, 84 (4) (1981), 409–422.Google Scholar
  68. [Pa64]
    F. Papangelou, Order convergence and topological completion of commutative lattice groups. Math. Annalen 155 (1964), 81–107.MathSciNetMATHCrossRefGoogle Scholar
  69. [PW89]
    J. R. Porter & R. G. Woods, Extensions and Absolutes of Hausdorff Spaces. Springer Verlag (1989); Berlin-Heidelberg-New York.Google Scholar
  70. [RW99]
    R. M. Raphael & R. G. Woods, The epimorphic hull of C(X). To appear.Google Scholar
  71. [Ri40]
    F. Riesz, Sur quelques notions fondamentales dans la théorie générale des opérations linéaires. Annals Math. 41 (1940), 174–206.MathSciNetCrossRefGoogle Scholar
  72. [SM99]
    N. Schwartz & J. J. Madden, Semi-Algebraic Function Rings and Reflectors of Partially Ordered Rings. Lecture Notes in Math. 1712 (1999), Springer Verlag; Berlin-Heidelberg- et. al.MATHGoogle Scholar
  73. [Si62]
    F. Sik, Über die Beziehungen zwischen eigenen Spitzen und minimalen Komponenten einer ℓ-Gruppe. Acta Math. Acad. Sci. Hungar. 13 (1962), 171–178.MathSciNetMATHCrossRefGoogle Scholar
  74. [St68]
    H. H. Storrer, Epimorphismen von kommutativen Ringen. Comm. Math. Helv. 43 (1968), 378–401.MathSciNetMATHCrossRefGoogle Scholar
  75. [U56]
    Y. Utumi, On quotient rings. Osaka Math. Jour. 8 (1956), 1–18.MathSciNetMATHGoogle Scholar
  76. [V69]
    A. I. Veksler, A new construction of the Dedekind completion for vector lattices and divisible ℓ-groups. Siberian Math. J. 10 (1969), 891–896.MathSciNetMATHCrossRefGoogle Scholar
  77. [VG72]
    A. I. Veksler & V. A. Geiler, Order and disjoint completeness of linear partially ordered spaces. Siberian Math. J. 13 (1972), 43–51.MathSciNetCrossRefGoogle Scholar
  78. [V84]
    J. Vermeer, The smallest basically disconnected preimage of a space. Topology and its Appl. 17 (1984), 217–232.MathSciNetCrossRefGoogle Scholar
  79. [W74]
    R. C. Walker, The Stone-Čech Compactification. Ergebnisse der Math. u. i. Grenzgeb. 83 (1974), Berlin-Heidelberg-New York.MATHCrossRefGoogle Scholar
  80. [Wi89]
    A. W. Wickstead, An intrinsic characterization of self-injective semiprime commutative rings. Proc. Royal Irish Acad., Section A, 90A (1) (1989), 117–124.MathSciNetGoogle Scholar

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© Springer Science+Business Media Dordrecht 2002

Authors and Affiliations

  • Jorge Martínez
    • 1
  1. 1.Department of MathematicsUniversity of FloridaGainesvilleUSA

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