  Consider a commutative monoid $(M,+,0)$ and a biadditive binary operation
$\mu \colon M \times M \to M$. We will show that under some additional general
assumptions, the operation $\mu$ is automatically both associative and
commutative. The main additional assumption is localizability of $\mu$, which
essentially means that a certain canonical order on $M$ is compatible with
adjoining some multiplicative inverses of elements of $M$. As an application we
show that a division ring $\mathbb F$ is commutative provided that for all $a
\in \mathbb F$ there exists a natural number $k$ such that $a-k$ is not a sum
of products of squares. This generalizes the classical theorem that every
archimedean ordered division ring is commutative to a more general class of
formally real division rings that do not necessarily allow for an archimedean
(total) order. Similar results about automatic associativity and commutativity
are well-known for special types of partially ordered extended rings
(``extended'' in the sense that neither associativity nor commutativity of the
multiplication is required by definition), namely in the uniformly bounded and
the lattice-ordered cases, i.e.~for (extended) $f$-rings. In these cases the
commutative monoid in question is the positive cone of the partially ordered
extended ring. We also discuss how these classical results can be obtained from
our main theorem.
