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Representations of $ l$-groups by almost-finite quotient maps


Author: Donald A. Chambless
Journal: Proc. Amer. Math. Soc. 28 (1971), 59-62
MSC: Primary 06.75
DOI: https://doi.org/10.1090/S0002-9939-1971-0274366-1
MathSciNet review: 0274366
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Abstract: There are numerous existing methods of embedding an archimedean lattice group into a space of extended-real-valued continuous functions defined on a topological space. In this paper the topological space is constructed using the hull-kernel topology and some prime subgroups of a certain extension of the given group, and the representing functions are described as certain quotient maps. The resulting representation essentially coincides with earlier representations given by B. Vulih and by S. J. Bernau.


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  • [1] I. Amemiya, A general spectral theory in semi-ordered linear spaces, J. Fac. Sci. Hokkaido Univ. Ser. I 12 (1953), 111-156. MR 15, 137. MR 0056853 (15:137d)
  • [2] B. Banaschewski, On lattice-ordered groups, Fund. Math. 55 (1964), 113-122. MR 29 #5930. MR 0168672 (29:5930)
  • [3] S. J. Bernau, Unique representation of Archimedean lattice groups and normal Archimedean lattice rings, Proc. London Math. Soc. (3) 15 (1965), 599-631. MR 32 #144. MR 0182661 (32:144)
  • [4] -, Orthocompletion of lattice groups, Proc. London Math. Soc. (3) 16 (1966), 107-130. MR 32 #5554. MR 0188113 (32:5554)
  • [5] G. Birkhoff, Lattice theory, 3rd ed., Amer. Math. Soc. Colloq. Publ., vol. 25, Amer. Math. Soc., Providence, R.I., 1967. MR 37 #2638. MR 0227053 (37:2638)
  • [6] P. Conrad, Introduction à la théorie des groupes réticulés, Secrétariat mathématique, Paris, 1967. MR 37 #1289.
  • [7] P. Conrad and D. McAlister, The completion of a lattice-ordered group, J. Austral Math. Soc. 9 (1969), 182-208. MR 0249340 (40:2585)
  • [8] L. Fuchs, Partially ordered algebraic systems, Pergamon Press, New York; Addison-Wiley, Reading, Mass., 1963. MR 30 #2090. MR 0171864 (30:2090)
  • [9] D. G. Johnson and J. E. Kist, Complemented ideals and extremally disconnected spaces, Arch. Math. 12 (1961), 349-354. MR 26 #6082. MR 0148575 (26:6082)
  • [10] -, Prime ideals in vector lattices, Canad. J. Math. 14 (1962), 517-528. MR 25 #2010. MR 0138566 (25:2010)
  • [11] L. V. Kantorovič, B. Z. Vulih and A. G. Pinsker, Functional analysis in partially ordered spaces, GITTL, Moscow, 1950. (Russian) MR 12, 340. MR 0038006 (12:340d)
  • [12] H. Nakano, Modern spectral theory, Maruzen, Tokyo, 1950. MR 12, 419. MR 0038564 (12:419f)
  • [13] D. Papert, A representation theory for lattice-groups, Proc. London Math. Soc. (3) 12 (1962), 100-120. MR 24 #A3217. MR 0133383 (24:A3217)
  • [14] M. H. Stone, A general theory of spectra. II, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 83-87. MR 2, 318. MR 0004092 (2:318a)
  • [15] B. Z. Vulih, Introduction to the theory of partially ordered spaces, Fizmatgiz, Moscow, 1961; English transl., Noordhoff, Groningen, 1967. MR 24 #A3494; MR 37 #121. MR 0224522 (37:121)
  • [16] K. Yosida, On vector lattice with a unit, Proc. Imp. Acad. Tokyo 17 (1941), 121-124. MR 3, 210. MR 0005795 (3:210a)
  • [17] -, On the representation of the vector lattice, Proc. Imp. Acad. Tokyo 18 (1942), 339-342. MR 7, 409. MR 0015378 (7:409f)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1971-0274366-1
Keywords: Representation of archimedean $ l$-groups, almost-finite continuous function, Stone topological space
Article copyright: © Copyright 1971 American Mathematical Society

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