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Each of the rings $\mathbf{Q}$, $\mathbf{Z}_p$, $\mathbf{Q}_p$, $\mathbf{R}$ admits a universal construction: the rationals are the field of fractions of the integers: $\mathbf{Q}:=\mathrm{Frac}(\mathbf{Z})$; the $p$-adic integers $\mathbf{Z}_p$ and $p$-adic numbers $\mathbf{Q}_p$ are the completion of the rings $\mathbf{Z}$ and $\mathbf{Q}$ with respect to the $p$-adic valuation; the ring $\mathbf{R}$ is the completion of $\mathbf{Q}$ with respect to the infinite valuation.

Some of these rings also have natural characterisations among one or other specific class of rings. For example, $\mathbf{R}$ is the terminal archimedean field, while $\mathbf{Q}$ is the initial characteristic zero field.

Are there universal constructions or characterisations for the tropical semiring?

To be more precise, there are four different semirings that are usually called "the" tropical semiring:

  1. the set $\mathbf{N}\cup\{\infty\}$ equipped with $\max$ as addition and $+$ as multiplication;
  2. the set $\mathbf{R}\cup\{\infty\}$ with the same operations;
  3. each of the above examples, but with $-\infty$ instead of $\infty$ and with $\min$ instead of $\max$ as the addition.

However, this last point does not matter: the semirings $\mathbf{N}\cup\{\infty\}$ and $\mathbf{N}\cup\{-\infty\}$ are canonically isomorphic, and similarly for the version involving $\mathbf{R}$. So it suffices to focus on $\mathbf{N}\cup\{\infty\}$ and $\mathbf{R}\cup\{\infty\}$ only.

The semirings $\mathbf{R}\cup\{\infty\}$ and $\mathbf{N}\cup\{\infty\}$ are both idempotent semirings, meaning that $a+a=a$ for each element $a$ in these rings. In addition, $\mathbf{R}\cup\{\infty\}$ carries a natural topology making addition and multiplication into continuous morphisms, turning it into a topological semiring, characterised as the unique one making the map $-\log:\mathbf{R}_{\geq0}\to\mathbf{R}\cup\{\infty\}$ into a homeomorphism. So does $\mathbf{N}\cup\{\infty\}$, carrying the order topology, which agrees with both the subspace topology with respect to $\mathbf{R}\cup\{\infty\}$ and the one-point compactification topology.

My specific questions are:

  1. Are there universal constructions of $\mathbf{N}\cup\{\infty\}$ and $\mathbf{R}\cup\{\infty\}$ as
    • semirings?
    • topological semirings?
  2. Are there characterisations of $\mathbf{N}\cup\{\infty\}$ and $\mathbf{R}\cup\{\infty\}$ as being universal with respect to some property among:
    • all semirings?
    • idempotent semirings?
    • topological semirings?

So far, there are the following universal characterisations of $\mathbf{N}\cup\{\infty\}$:

  • Schweber. The initial idempotent semiring with a nonzero additively indecomposable annihilator.
  • Dave. The semiring of ideals of any discrete valuation ring.
Emily
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    Here's an admittedly rather trivial observation re: the $\mathbb{N}\cup{\infty}$ version: Given a magma $(A;\star)$, say that an element $\alpha\in A$ is an unsplittable annihilator iff $\alpha\star b=b\star\alpha=\alpha$ for all $b\in A$ but if $b,c\not=\alpha$ then $b\star c\not=\alpha$. Then $\mathfrak{N}=(\mathbb{N}\cup{\infty}; \max,+,0,1,\infty)$ is the initial object in the category of additively idempotent, additively-and-multiplicatively unital semirigs with distinguished nonzero non-annihilator and distinguished unsplittable annihilator. (This is actually a bit overkill.) – Noah Schweber Sep 09 '21 at 17:39
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    @NoahSchweber If one allows some imprecision, then I think this becomes a rather nice slogan: "$\mathbf{N}\cup{\infty}$ is the initial idempotent semiring with a nonzero indecomposable annihilator"! – Emily Sep 09 '21 at 22:55
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    Crossposted: link. – Emily Sep 13 '21 at 01:51
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    A minor note: The order topology on $\mathbb N\cup{\infty}$ is the same as the one-point compactification topology. (Every point is open except $\infty$, and its neighbourhoods are the cofinite sets.) – Toby Bartels Sep 13 '21 at 02:39
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    @TobyBartels Thanks! – Emily Sep 13 '21 at 06:26

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