Associativity and Commutativity of Partially Ordered Rings

Abstract: 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.