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

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Part of the book series: Developments in Mathematics ((DEVM,volume 7))

Abstract

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.

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References

  1. F. W. Anderson, Lattice-ordered rings of quotients. Canad. Jour. Math. 17 (1965), 434–448.

    Article  MATH  Google Scholar 

  2. M. Anderson & T. Feil, Lattice-Ordered Groups, an Introduction. (1988) Reidel Texts in the Math. Sci.; Kluwer, Dordrecht.

    Book  MATH  Google Scholar 

  3. E. R. Aron, Embedding lattice-ordered algebras in uniformly closed algebras. (1971) Thesis, University of Rochester.

    Google Scholar 

  4. E. R. Aron & A. W. Hager, Convex vector lattices and ℓ-algebras. Topology and its Appl. 12 (1981), 1–10.

    Article  MathSciNet  MATH  Google Scholar 

  5. 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. 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.

    Article  MathSciNet  Google Scholar 

  7. R. N. Ball & A. W. Hager, Epicomplete archimedean ℓ-groups and vector lattices. Trans. AMS 322 (No. 2) (1990), 459–478.

    MathSciNet  MATH  Google Scholar 

  8. 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.

    Article  MathSciNet  MATH  Google Scholar 

  9. R. N. Ball & A. W. Hager, Algebraic extensions of an archimedean latticeordered group, I. J. of Pure & Appl. Algebra 85 (1993), 1–20.

    Article  MathSciNet  MATH  Google Scholar 

  10. R. N. Ball & A. W. Hager, Algebraic extensions of an archimedean latticeordered group, II. J. of Pure & Appl. Algebra 138 (1999), 197–204.

    Article  MathSciNet  MATH  Google Scholar 

  11. 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.

    Article  MathSciNet  MATH  Google Scholar 

  12. B. Banaschewski, Maximal rings of quotients of semi-simple commutative rings. Archiv. Math. XVI (1965), 414–420.

    Article  MathSciNet  Google Scholar 

  13. S. J. Bernau, The lateral completion of a lattice ordered group. J. Austral. Math. Soc. 19 (1975), 263–289.

    Article  MathSciNet  MATH  Google Scholar 

  14. S. J. Bernau, Lateral and Dedekind completion of archimedean lattice groups. J. London Math. Soc. 12 (1975/76), 320–322.

    Article  MathSciNet  Google Scholar 

  15. 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. G. Birkhoff, Lattice Theory (3rd Ed.) AMS Colloq. Publ. XXV (1967), Providence, RI.

    MATH  Google Scholar 

  17. R. D. Bleier, The SP-hull of a lattice-ordered group. Canad. Jour. Math. XXVI, No. 4 (1974), 866–878.

    Article  MathSciNet  Google Scholar 

  18. D. Chambless, The Representation and Structure of Lattice-Ordered Groups an f-Rings. Tulane University Dissertation (1971), New Orleans.

    Google Scholar 

  19. P. F. Conrad, The lateral completion of a lattice-ordered group. Proc. London Math. Soc. 3rd Series XIX (July 1969), 444–480.

    Article  MathSciNet  Google Scholar 

  20. P. F. Conrad, The essential closure of an archimedean lattice-ordered group. Duke Math. Jour. 38 (1971), 151–160.

    Article  MathSciNet  MATH  Google Scholar 

  21. P. F. Conrad, The hulls of representable ℓ-groups and f-rings. J. Austral. Math. Soc. 26 (1973), 385–415.

    Article  MathSciNet  Google Scholar 

  22. P. F. Conrad & D. McAlister, The completion of a lattice-ordered group. J. Austral. Math. Soc. 9 (1969), 182–208.

    Article  MathSciNet  MATH  Google Scholar 

  23. M. R. Darnel, The Theory of Lattice-Ordered Groups. Pure & Appl. Math. 187, Marcel Dekker (1995); Basel-Hong Kong-New York.

    Google Scholar 

  24. 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.

    Article  MathSciNet  MATH  Google Scholar 

  25. C. J. Everett, Sequence completions of lattice modules. Duke Math. J. 11 (1944), 109–119.

    Article  MathSciNet  MATH  Google Scholar 

  26. N. Fine, L. Gillman & J. Lambek, Rings of Quotients of Rings of Functions. (1965) McGill University.

    Google Scholar 

  27. L. Fuchs, Partially Ordered Algebraic Systems. (1963) Pergamon Press, Oxford-New York-London-Paris.

    MATH  Google Scholar 

  28. L. Gillman & M. Jerison, Rings of Continuous Functions. Grad. Texts in Math. 43, Springer Verlag (1976); Berlin-Heidelberg-New York.

    Google Scholar 

  29. A. M. Gleason, Projective topological spaces. Illinois J. Math. 7 (1958), 482–489.

    MathSciNet  Google Scholar 

  30. V. Glivenko, Bull. Acad. Sci. Belg. 15 (1929), 183–88.

    MATH  Google Scholar 

  31. 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. 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.

    MathSciNet  Google Scholar 

  33. 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. A. W. Hager & J. Martínez, Functorial rings of quotients, II. Forum Math. 6 (1994), 597–616.

    Article  MathSciNet  MATH  Google Scholar 

  35. A. W. Hager & J. Martinez, Maximum monoreflections. Appl. Categ. Struc. 2 (1994), 315–329.

    Article  MathSciNet  MATH  Google Scholar 

  36. 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. A. W. Hager & J. Martinez, Temple U. Ethiopian J. Sci. (1996), 73–84.

    Google Scholar 

  38. 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. A. W. Hager & J. Martínez, Pushout-invariant extensions and monoreflections. J. of Pure and Appl. Alg. 129 (1998), 263–295.

    Article  MATH  Google Scholar 

  40. A. W. Hager & J. Martínez, Singular archimedean lattice-ordered groups. Alg. Universalis 40 (1998), 119–147.

    Article  MATH  Google Scholar 

  41. A. W. Hager & J. Martínez, More on the laterally o-complete reflection of an archimedean lattice-ordered group. Order 15 (1999), 247–260.

    Article  Google Scholar 

  42. A. W. Hager & J. Martínez, Hulls for various kinds of a-completeness in archimedean lattice-ordered groups. Order 16 (1999), 89–103.

    Article  MathSciNet  MATH  Google Scholar 

  43. 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.

    MathSciNet  MATH  Google Scholar 

  44. A. W. Hager & J. Martínez, Maximum monoreflectionslections and essential extensions. Appl. Categ. Struc. 9 (2001), 517–523.

    Article  MATH  Google Scholar 

  45. 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. A. W. Hager & J. Martínez, The ring of a-quotients. To appear, Algebra Universalis.

    Google Scholar 

  47. A. W. Hager & J. Martinez, On strong a-regularity. Work in progress.

    Google Scholar 

  48. 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. A. W. Hager & J. Martínez, The projectable and regular hulls of a semiprime ring. Preprint.

    Google Scholar 

  51. A. W. Hager & J. Martinea, The regular reflection of a semiprime f-ring. Work in progress.

    Google Scholar 

  52. A. W. Hager & L. C. Robertson, Representing and ringifying a Riesz space. Symp. Math. 21 (1977), 411–431.

    MathSciNet  Google Scholar 

  53. A. W. Hager & L. C. Robertson, Extremal units in an archimedean Riesz space. Rend. Sem. Mat. Univ. Padova 59 (1978), 97–115.

    MathSciNet  MATH  Google Scholar 

  54. A. W. Hager & L. C. Robertson, On the embedding into a ring of an archimedean ℓ-group. Canad. J. Math. 31 (1979), 1–8.

    Article  MathSciNet  MATH  Google Scholar 

  55. M. Henriksen, J. R. Isbell & D. G. Johnson, Residue class fields of latticeordered algebras. Fund. Math. 50 (1961), 107–117.

    MathSciNet  MATH  Google Scholar 

  56. M. Henriksen & D. G. Johnson, On the structure of a class of lattice-ordered algebras. Fund. Math. 50 (1961), 73–94.

    MathSciNet  MATH  Google Scholar 

  57. M. Henriksen, J. Vermeer & R. G. Woods, The quasi F-cover of Tychonoff spaces. Trans. AMS 303 (1987), 779–803.

    MathSciNet  MATH  Google Scholar 

  58. M. Henriksen, J. Vermeer & R. G. Woods, Wallman covers of compact spaces. Diss. Math. CCLXXX (1989), Warsaw.

    Google Scholar 

  59. H. Herrlich & G. E. Strecker, Category Theory. Sigma Series in Pure Math. 1 (1979), Heldermann Verlag, Berlin.

    MATH  Google Scholar 

  60. C. B. Huismans & B. de Pagter, Ideal theory in f-algebras. Trans. AMS 269 (No. 1) (January, 1982), 225–245.

    Google Scholar 

  61. C. B. Huijsmans & B. de Pagter, Maximal d-ideals in a Riesz space. Canad. Jour. Math. 35 (1983), 1010–1029.

    Article  MATH  Google Scholar 

  62. J. Lambek, Lectures on Rings and Modules. (3rd Ed.) (1986) Chelsea Publ. Co., New York.

    Google Scholar 

  63. W. A. J. Luxemburg & A. C. Zaanen, Riesz Spaces, I. (1971) North Holland, Amsterdam.

    Google Scholar 

  64. J. J. Madden & J. Vermeer, Epicomplete archimedean ℓ-groups via a localic Yosida theorem. J. of Pure & Appl. Alg. 68 (1990), 243–252.

    Article  MathSciNet  MATH  Google Scholar 

  65. J. Martinez, The maximal ring of quotients of an f-ring. Alg. Universalis 33 (1995), 335–369.

    Google Scholar 

  66. J. Martfnez, Polar functions, I: The summand-inducing hull of an archimedean lattice-ordered group with unit. In these proceedings.

    Google Scholar 

  67. 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. F. Papangelou, Order convergence and topological completion of commutative lattice groups. Math. Annalen 155 (1964), 81–107.

    Article  MathSciNet  MATH  Google Scholar 

  69. J. R. Porter & R. G. Woods, Extensions and Absolutes of Hausdorff Spaces. Springer Verlag (1989); Berlin-Heidelberg-New York.

    Google Scholar 

  70. R. M. Raphael & R. G. Woods, The epimorphic hull of C(X). To appear.

    Google Scholar 

  71. F. Riesz, Sur quelques notions fondamentales dans la théorie générale des opérations linéaires. Annals Math. 41 (1940), 174–206.

    Article  MathSciNet  Google Scholar 

  72. 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.

    MATH  Google Scholar 

  73. F. Sik, Über die Beziehungen zwischen eigenen Spitzen und minimalen Komponenten einer ℓ-Gruppe. Acta Math. Acad. Sci. Hungar. 13 (1962), 171–178.

    Article  MathSciNet  MATH  Google Scholar 

  74. H. H. Storrer, Epimorphismen von kommutativen Ringen. Comm. Math. Helv. 43 (1968), 378–401.

    Article  MathSciNet  MATH  Google Scholar 

  75. Y. Utumi, On quotient rings. Osaka Math. Jour. 8 (1956), 1–18.

    MathSciNet  MATH  Google Scholar 

  76. A. I. Veksler, A new construction of the Dedekind completion for vector lattices and divisible ℓ-groups. Siberian Math. J. 10 (1969), 891–896.

    Article  MathSciNet  MATH  Google Scholar 

  77. A. I. Veksler & V. A. Geiler, Order and disjoint completeness of linear partially ordered spaces. Siberian Math. J. 13 (1972), 43–51.

    Article  MathSciNet  Google Scholar 

  78. J. Vermeer, The smallest basically disconnected preimage of a space. Topology and its Appl. 17 (1984), 217–232.

    Article  MathSciNet  Google Scholar 

  79. R. C. Walker, The Stone-Čech Compactification. Ergebnisse der Math. u. i. Grenzgeb. 83 (1974), Berlin-Heidelberg-New York.

    Book  MATH  Google Scholar 

  80. A. W. Wickstead, An intrinsic characterization of self-injective semiprime commutative rings. Proc. Royal Irish Acad., Section A, 90A (1) (1989), 117–124.

    MathSciNet  Google Scholar 

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Martínez, J. (2002). Hull Classses of Archimedean Lattice-Ordered Groups with Unit: A Survey. In: Martínez, J. (eds) Ordered Algebraic Structures. Developments in Mathematics, vol 7. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3627-4_5

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  • DOI: https://doi.org/10.1007/978-1-4757-3627-4_5

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