TITLE: Is the reals the smallest connected ordered topological ring?
QUESTION [10 upvotes]: The real numbers is a locally compact Tychonoff connected complete ordered topological field. I am looking at minimal collections of adjectives that can characterize the reals. The one often used to define the reals is that it is (Dedekind- or Cauchy-) complete ordered field.
Consider the real numbers among other ordered topological rings. I am wondering if the real numbers is the smallest connected ordered topological ring? Here "smallest" means that it embeds into every other connected ordered topological ring. If this is not true, do you have some minimal collection of additional adjectives (advoiding the word "complete") that can characterize the real numbers among ordered topological rings? "Minimal" means that if you drop an adjective then you there are other ordered topological rings that also fulfil your definition.

REPLY [13 votes]: The answer to the original question is positive, and even more is true.

Proposition: $(\mathbb R,+,\le)$ is up to isomorphism the unique nontrivial connected ordered group.

Proof:
Let $(G,+,\le)$ be such a group (not necessarily abelian despite the notation). First, observe that $G$ is archimedean, i.e., for every $a,b>0$, there is a natural number $n$ such that $b\le na$. This follows from the fact that
$$\{b:\exists n\in\mathbb N\,(-na\le b\le na)\}$$
is a nontrivial convex subgroup, hence clopen.
Second, by a well-known theorem of Hölder, every archimedean group is isomorphic to a subgroup of $\mathbb R$. The argument is as follows. We fix a positive element $a$. For any $b\in G$ and $n\in\mathbb N$, there is a unique $b_n\in\mathbb Z$ such that $b_na\le nb<(b_n+1)a$. Then it is easy to show that
$$f(b):=\lim_{n\to\infty}\frac{b_n}n$$
exists, and that $f$ is an ordered group embedding of $G$ into $\mathbb R$.
Now, $\mathbb R$ has no nontrivial proper connected subgroup: if $a\in\mathbb R\smallsetminus G$, then $(-\infty,a)\cap G$ and $(a,+\infty)\cap G$ form an open partition of $G$. (By the way, in general an ordered space is connected iff it is densely ordered and Dedekind complete.)
Since there is only one way how to expand $(\mathbb R,+,\le,1)$ to an ordered ring, we obtain

Corollary: $(\mathbb R,+,\cdot,\le)$ is up to isomorphism the unique nontrivial connected ordered ring.