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arXiv:2303.03627v4 Announce Type: replace
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 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 rings, namely in the uniformly bounded and the lattice-ordered cases, i.e. for $f$-algebras. In these cases the commutative monoid in question is the positive cone of the ordered ring. We also discuss how these classical results can be obtained from the main theorem.