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 $\mathbb{Q}$)?