Well-ordered commutative semirings

Question

I am interested in the characterization of the most important types of numbers from an axiomatic viewpoint. For example, every complete ordered field is isomorphic to the field of real numbers. In this sense, being a complete ordered field characterizes the real numbers. I am now looking at the much simpler set of positive integers. It is a well-ordered commutative semiring. Does this property characterize them? In other words, are there any well-ordered commutative semirings other than the positive integers?

Answer 1

No, that's not nearly enough for an exact characterization. A good counterexample to consider is the set $\mathbb{N}[x]$ of polynomials in a single variable with nonnegative integer coefficients, with addition and multiplication given in the obvious way and ordered by setting $$f<g\quad\iff\quad f(x)<g(x)\mbox{ for all sufficiently large $x$}.$$ This is a countable commutative well-ordered semiring; moreover, the ordering "plays nicely" with the algebraic structure (a la ordered fields or ordered groups) and we have a cancellation property (which distinguishes this from Brian Moehring's comment above).

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That said, here are a couple characterizations which do work:

• $(\mathbb{N};0,1,+,\times,<)$ is the unique well-ordered commutative unital semiring satisfying the "difference condition" that for each $x,y$ there is some $z$ such that either $x+z=y$ or $y+z=x$.

• $(\mathbb{N};0,1,+,\times,<)$ is the unique well-ordered commutative unital semiring in which every nonzero element has a predecessor.

• $(\mathbb{N};0,1,+,\times)$ is the initial object in the category of unital semirings (and we can add ordering to this if we want).