Archimedean-like classes of lattice-ordered groups

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.