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Transactions of the American Mathematical Society

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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.

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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

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