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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Archimedean-like classes of lattice-ordered groups
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by Jorge Martinez PDF
Trans. Amer. Math. Soc. 186 (1973), 33-49 Request permission

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.
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Additional Information
  • © Copyright 1973 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 186 (1973), 33-49
  • MSC: Primary 06A55
  • DOI: https://doi.org/10.1090/S0002-9947-1973-0332614-X
  • MathSciNet review: 0332614