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Question. Is there a suitable notion of the Laurent series ring $\mathbb{F}_1((t))$ and power series ring $\mathbb{F}_1[[t]]$ in some framework for the field with one element $\mathbb{F}_1$?

For example, the category of monoids is one way to 'model' algebras over field with one element or $\mathbb{F}_1$, due to A. Deitmar building on the ideas of K. Kato. The $\mathbb{F}_1$-algebras or affine $\mathbb{F}_1$-schemes are monoids and the base change functor to $\mathbb{Z}$-algebras sends a monoid $A$ to the monoid algebra $\mathbb{Z}[A]$. There is a notion of 'monoidal space' analogous to ringed space and a notion of $\mathbb{F}_1$-scheme that is a gluing of affine $\mathbb{F}_1$-schemes.

In this category, we can think of $\mathbb{F}_1[t]$ as $\mathbb{N}$, for instance by thinking of it as the free monoid on one generator.

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    $\begingroup$ If $\mathbb{F}_1[t]$ is $\mathbb{N}$, then maybe $\mathbb{F}_1[[t]]$ is the profinite completion of $\mathbb{N}$, and Laurent series are obtained by adjoining an inverse of $1$. $\endgroup$ Sep 15, 2015 at 17:24
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    $\begingroup$ §3.3 of Tobias Dyckerhoff's notes makes a suggestion that finitary $\mathbb{F}_1\left[\left[t\right]\right]$-modules (up to isomorphism) are indexed by partitions. What $\mathbb{F}_1\left[\left[t\right]\right]$ is itself? I don't know. $\endgroup$ Sep 15, 2015 at 18:40
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    $\begingroup$ Dyckerhoff's paper arxiv.org/abs/1505.06940 says that finite $\mathbb F_1[[t]]$-modules should be considered as an $\mathbb F_1$ vector space (i.e. finite set with distinguished element $*$) together with a nilpotent endomorphism. (I guess this is a different paper from the one @darijgrinberg was looking at; the parts of the partition arise here as lengths of maximal paths to $*$.) $\endgroup$ Dec 30, 2015 at 21:02
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    $\begingroup$ @HughThomas: Oops, the notes I was referencing are no longer on his website. They are archived at web.archive.org/web/20150601115236/http://www.math.uni-bonn.de/… . $\endgroup$ Dec 30, 2015 at 22:16

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