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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Archimedean-like classes of lattice-ordered groups

Author: Jorge Martinez
Journal: Trans. Amer. Math. Soc. 186 (1973), 33-49
MSC: Primary 06A55
MathSciNet review: 0332614
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Abstract: Suppose $ \mathcal{C}$ denotes a class of totally ordered groups closed under taking subgroups and quotients by o-homomorphisms. We study the following classes: (1) $ {\text{Res}}(\mathcal{C})$, the class of all lattice-ordered groups which are subdirect products of groups in $ \mathcal{C}$; (2) $ {\text{Hyp}}(\mathcal{C})$, the class of lattice-ordered groups in $ {\text{Res}}(\mathcal{C})$ having all their l-homomorphic images in $ {\text{Res}}(\mathcal{C})$; Para $ (\mathcal{C})$, the class of lattice-ordered groups having all their principal convex l-subgroups in $ {\text{Res}}(\mathcal{C})$. If $ \mathcal{C}$ is the class of archimedean totally ordered groups then Para $ (\mathcal{C})$ is the class of archimedean lattice-ordered groups, $ {\text{Res}}(\mathcal{C})$ is the class of subdirect products of reals, and $ {\text{Hyp}}(\mathcal{C})$ consists of all the hyper archimedean lattice-ordered groups.

We show that under an extra (mild) hypothesis, any given representable lattice-ordered group has a unique largest convex l-subgroup in $ {\text{Hyp}}(\mathcal{C})$; this socalled hyper- $ \mathcal{C}$-kernel is a characteristic subgroup. We consider several examples, and investigate properties of the hyper- $ \mathcal{C}$-kernels.

For any class $ \mathcal{C}$ as above we show that the free lattice-ordered group on a set X in the variety generated by $ \mathcal{C}$ is always in $ {\text{Res}}(\mathcal{C})$. We also prove that $ {\text{Res}}(\mathcal{C})$ has free products.

References [Enhancements On Off] (What's this?)

  • [1] R. Bleier, Dissertation, Tulane University, New Orleans, La., 1971.
  • [2] P. M. Cohn, Universal algebra, Harper & Row, Publishers, New York-London, 1965. MR 0175948
  • [3] Paul Conrad, A characterization of lattice-ordered groups by their convex𝑙-subgroups, J. Austral. Math. Soc. 7 (1967), 145–159. MR 0214521
  • [4] -, Lattice-ordered groups, Tulane University, New Orleans, La., 1970.
  • [5] Paul Conrad and Donald McAlister, The completion of a lattice ordered group, J. Austral. Math. Soc. 9 (1969), 182–208. MR 0249340
  • [6] L. Fuchs, Partially ordered algebraic systems, Pergamon Press, Oxford-London-New York-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass.-Palo Alto, Calif.-London, 1963. MR 0171864
  • [7] Jorge Martinez, Free products of abelian 𝑙-groups, Czechoslovak Math. J. 23(98) (1973), 349–361. MR 0371770
  • [8] Hanna Neumann, Varieties of groups, Springer-Verlag New York, Inc., New York, 1967. MR 0215899
  • [9] Elliot Carl Weinberg, Free lattice-ordered abelian groups, Math. Ann. 151 (1963), 187–199. MR 0153759
  • [10] Samuel Wolfenstein, Valeurs normales dans un groupe réticulé, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 44 (1968), 337–342 (French, with Italian summary). MR 0234887

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Keywords: Closed class of o-groups, residually- $ \mathcal{C}$ l-groups, hyper- $ \mathcal{C}$ l-groups, para- $ \mathcal{C}$ l-groups, hyper- $ \mathcal{C}$ kernel, c-archimedean l-groups, $ \mathcal{C}$ radical
Article copyright: © Copyright 1973 American Mathematical Society