Archimedeanlike classes of latticeordered groups
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 by Jorge Martinez PDF
 Trans. Amer. Math. Soc. 186 (1973), 3349 Request permission
Abstract:
Suppose $\mathcal {C}$ denotes a class of totally ordered groups closed under taking subgroups and quotients by ohomomorphisms. We study the following classes: (1) ${\text {Res}}(\mathcal {C})$, the class of all latticeordered groups which are subdirect products of groups in $\mathcal {C}$; (2) ${\text {Hyp}}(\mathcal {C})$, the class of latticeordered groups in ${\text {Res}}(\mathcal {C})$ having all their lhomomorphic images in ${\text {Res}}(\mathcal {C})$; Para $(\mathcal {C})$, the class of latticeordered groups having all their principal convex lsubgroups 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 latticeordered groups, ${\text {Res}}(\mathcal {C})$ is the class of subdirect products of reals, and ${\text {Hyp}}(\mathcal {C})$ consists of all the hyper archimedean latticeordered groups. We show that under an extra (mild) hypothesis, any given representable latticeordered group has a unique largest convex lsubgroup 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 latticeordered 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

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Additional Information
 © Copyright 1973 American Mathematical Society
 Journal: Trans. Amer. Math. Soc. 186 (1973), 3349
 MSC: Primary 06A55
 DOI: https://doi.org/10.1090/S0002994719730332614X
 MathSciNet review: 0332614