Let $\Bbbk$ be a field; I am interested in the following ring (which I suspect is a field). Its elements are formal expressions that look like

$$ \sum_{n=0}^{\infty} a_n x^{b_n} $$

where $a_n\in \Bbbk$ and $b_n\in \mathbb{R}$, with $b_n$ strictly increasing, and $\lim_{n\to\infty} b_n = \infty$. (Technically, we should quotient these by some relation allowing us to insert and remove terms where $a_n=0$. Or we could require all the $a_n$'s to be nonzero, but then we'd have to include finite sums as well. Or we could represent them by functions $a : \mathbb{R} \to \Bbbk$ assigning a coefficient to each exponent, whose support is left-finite.) We can add and multiply these expressions in fairly evident ways; the condition on $b_n$ ensures that multiplication works (i.e. the resulting set of exponents can again be enumerated with order type $\omega$ and limit $\infty$).

This ring of power-series-like-objects is closely related to some others. Specifically, if $\Bbbk=\mathbb{R}$ then it contains the Levi-Civita field as the elements for which each $b_n\in\mathbb{Q}$, while it is contained in the Hahn series field $\Bbbk[[x^{\mathbb{R}}]]$. Note that the set of all Hahn series with order type $\omega$ is not closed under multiplication; this is a natural subset thereof that is. I believe that it is also the set of Hahn series with order type $\omega$ that converge to themselves in the valuation topology of the Hahn series field, and also that it is the closure of the field $\Bbbk(x^{\mathbb{R}})$ of generalized rational functions inside the Hahn series field, and the Cauchy completion of $\Bbbk(x^{\mathbb{R}})$ in its valuation uniformity.

Does this field have a standard name and/or a notation? Is it an instance of some more general construction (e.g. replacing $\mathbb{R}$ by something more general, which would presumably then also include the Levi-Civita field as the case of $\mathbb{Q}$)?

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    $\begingroup$ This is the completion of $\mathbb{k}(x^{\mathbb{R}})$ as a valued field, i.e. the unique up to isomorphism dense valued field extension without proper dense valued field extension. $\endgroup$
    – nombre
    Jul 6 '19 at 10:29
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    $\begingroup$ It is a field because the multiplication is well-defined commutative and the geometric series implies $1+\sum_{n=1}^\infty a_n x^{b_n}, 0 < b_n <b_{n+1} \to \infty$ has an inverse. $\endgroup$
    – reuns
    Jul 6 '19 at 22:05

I think your ring looks similar to the Novikov ring (see topology papers).

  • $\begingroup$ Following Google to en.wikipedia.org/wiki/Novikov_ring and thence by an external link back to MO, I found mathoverflow.net/a/13226/49 where my ring is called the "universal Novikov field" (hence I guess it is a field!). Going back to Google, I see that this terminology seems to be common. Unfortunately the most common notation for it is $\Lambda$ or $\Lambda(\Bbbk)$, which works in the context of a particular paper but can't really be a global notation (like $\Bbbk[[x]]$) -- for one thing, it clashes with the standard notation for exterior algebras. But a name helps! $\endgroup$ Jul 6 '19 at 15:48
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    $\begingroup$ @MikeShulman: Funny: 4 reviewers out of 4 voted to delete this post. It was only you accepting it as an answer that stopped the review process and saved the post. $\endgroup$
    – Alex M.
    Jul 6 '19 at 17:38
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    $\begingroup$ @AlexM. Yes, that is funny. I would argue that those reviewers were being too trigger-happy. Along the lines of meta.stackexchange.com/q/225370, I think this is an answer, and it did contain enough information to lead me to what I wanted to know, even if it would have been nice for it to contain more information to save me the trouble of Googling and following links. $\endgroup$ Jul 6 '19 at 18:00

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