# An equivariant Hahn Embedding Theorem?

The Hahn Embedding Theorem asserts that for any (linearly) ordered abelian group $\Lambda$, there exists a linearly ordered indexing set $\Omega$ such that $\Lambda$ admits an order-preserving group embedding in the subgroup $\Lambda_\Omega$ of $\mathbb{R}^\Omega$ consisting of those entries with well-ordered support. ($\Lambda_\Omega$ is itself an ordered abelian group.)

This result has been extended in various directions, for example to lattice-ordered groups, and to the case where the given group is not abelian.

What I would like to know is if there is an equivariant version of this result. Let $\mathrm{Aut}^+(\Lambda)$ denote the group of order-preserving group automorphisms ($o$-automorphisms) of the ordered abelian group $\Lambda$.

Does there exist a linearly ordered set $\Omega$, an $o$-embedding $h:\Lambda\to\Lambda_\Omega$ and an embedding $\eta:\mathrm{Aut}^+(\Lambda)\to \mathrm{Aut}^+(\Lambda_\Omega)$ ($\gamma\mapsto\eta_\gamma$) such that $$\eta_\gamma\cdot h=h\cdot \gamma?$$ In addition, can $\Omega$ be chosen so that $\mathrm{Aut}^+(\Lambda_\Omega)$ acts transitively on the positive elements of $\Lambda_\Omega$?

Looking at A. H. Clifford's proof (Proc. Amer. Math. Soc., 5 (1954) 860-863), it seems plausible that extending $o$-automorphisms of $\Lambda$ in the required way might be possible, but I may well be overlooking some additional hypothesis. There are some promising-sounding papers, such as M. Droste and R. Goebel "The Automorphism Groups of Hahn Groups", but unfortunately I don't have access to this. (It sounds from the review on mathscinet that it doesn't address the question above.)

In any case, I imagine that questions like this have been asked -- and answered -- before, hence the reference-request tag.

EDIT (9 July 2014): Here's an outline of how I think one could answer the main question above in the affirmative.

The Hahn Embedding Theorem already gives an embedding of an arbitrary $\Lambda$ in $\Lambda_{\Omega}$ where $\Omega$ is (isomorphic to) the linearly ordered set of principal convex subgroups of $\Lambda$, that is, those convex subgroups generated (as a convex subgroup) by a single element. Let's write $\alpha_g$ for a typical element of $\mathrm{Aut}^+(\Lambda)$. Each $\alpha_g\in\mathrm{Aut}^+(\Lambda)$ induces an order-preserving permutation $\rho_g$ of the set of principal convex subgroups, and thus of $\Omega$. This gives an $o$-automorphism $\sigma_g$ of $\Lambda_\Omega$ that has the "same" effect on the principal convex subgroups of $\Lambda_\Omega$ as $\alpha_g$ has on $\Lambda$: that is, $h\cdot\alpha_g(\Lambda_0)$ and $\sigma_g\cdot h(\Lambda_0)$ span the same convex subgroup of $\Lambda_\Omega$, for convex subgroups $\Lambda_0$ of $\Lambda$.

Next, we can rescale the $\omega$th entry of an element of $\Lambda_\Omega$ (by multiplying by a positive real number $\kappa_\omega$) for $\omega\in\Omega$, and add suitable multiples of the $\omega$th entry to subsequent entries in such a way that we obtain an $o$-automorphism $\bar{\alpha}_g=\tau_g\cdot\kappa_g\cdot\sigma_g$ of $\Lambda_\Omega$ such that $h\cdot\alpha_g=\bar{\alpha}_g\cdot h$. Here $\kappa_g=(\kappa_\omega)$ is the coordinatewise rescaling map, and $\tau_g$ is a `unitary' map, which can be thought of as an infinite-dimensional analogue of an upper triangular matrix with real entries and diagonal entries equal to one, which records the effect of adding multiples of the $\omega$th entry to the $\omega'$th entry ($\omega<\omega'$).

But my suspicion remains that this particular wheel was invented a long time ago.