Tarski’s linear-order commutation problem

Tarski posed the problem in 1956: an isomorphism between finite sums with possibly different positive coefficients should force the two underlying orders to commute under addition.

Known results

• Tarski proved the implication when $n=l$ or $m=k$.
• C. C. Chang extended this to $n\leq l$ and $m\leq k$.
• The remaining mixed coefficient inequalities were open according to the preprint.

August 24, 2026 affirmative preprint

On August 24, 2026, the arXiv preprint The Additive Arithmetic of Linear Orders claimed that the answer is affirmative in general, citing its Theorem 7.6 and a Euclidean-algorithm argument for arbitrary positive coefficients. This would settle the problem for all linear orders; the source also says that extension to arbitrary ordinal algebras is future work. The claim is unverified.

Current status (as of August 2026): The problem is claimed solved for all linear orders by an affirmative arXiv preprint, but independent verification of the proof is not recorded.