A Priestley-type Method for Generating Free l-Groups

  • Néstor G. Martínez
  • Alejandro Petrovich
Part of the Developments in Mathematics book series (DEVM, volume 7)


A Priestley-type topological representation is developed for the class of partially ordered groups which are p. o. subgroups of lattice-ordered groups. The appropriated analog of the spectrum of prime lattice filters is the class of increasing subsets P satisfying abP and cdP imply adP or cbP. In developing this representation, we give new embedding theorems for these groups. In particular, we give a necessary and sufficient condition for a p. o. group in this class to be embedded in an l-group of sets in such a way that the embedding preserves meets and joins that already exist in the group. Our construction gives also an alternative and very natural way to obtain the free l-group generated by a p. o. group in this class.


Distributive Lattice Positive Cone Opposite Inclusion Complete Family Priestley Space 
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  1. [1]
    A. Bigard, K. Keimel and S. Wolfenstein, Groupes et Anneaux Réticulés. Lecture Notes in Math. 608 (1977), Springer-Verlag, New York.zbMATHGoogle Scholar
  2. [2]
    P. F. Conrad, Right-ordered groups. Michigan Math. J. 6 (1959), 267–275.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    P. F. Conrad, Free lattice-ordered groups. J. Algebra 16 (1970), 191–203.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    B. A. Davey and H. A. Priestley, Introduction to Lattices and Order. (1990) Cambridge University Press.zbMATHGoogle Scholar
  5. [5]
    A. M. W. Glass and W. C. Holland (eds), Lattice-Ordered Groups, Advances and Techniques. (1989) Kluwer Acad. Publ., Dordrecht.zbMATHGoogle Scholar
  6. [6]
    W. C. Holland, The lattice ordered group of automorphisms of an ordered set. Michigan Math. J. 10 (1963), 399–408.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    N. G. Martínez, A topological duality for lattice-ordered algebraic structures including l-groups. Alg. Univ. 31 (1994), 516–541.Google Scholar
  8. [8]
    N. G. Martínez, Spectra and embeddings theorems for ordered groups. Proc. of the IX Latin American Symposium on Mathematical Logic; Notas de Lógica Matemática 39 Part 2 (1994), INMABB-CONICET, UNS, Bahía Blanca, 131–143.Google Scholar
  9. [9]
    P. Ribenboim, Théorie des groupes ordonnés. Monografías de Matemática (1959), Instituto de Matemática, UNS, Bahía Blanca.Google Scholar
  10. [10]
    C. Tsinakis, Personal communication. Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2002

Authors and Affiliations

  • Néstor G. Martínez
    • 1
  • Alejandro Petrovich
    • 1
  1. 1.Departamento de MatemdticaUniversidad de Buenos AiresBuenos AiresArgentina

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