MathDBTarski’s linear-order commutation problem
Tarski’s linear-order commutation problem
For every pair of linear orders and and every , if the ordered sums satisfy , where denotes the ordered sum of copies of , then .
Progress summary
A new preprint gives an affirmative answer to this 70-year-old question about when two ordered additions can be interchanged.
Tarski posed the linear-order commutation problem in 1956. It asks whether the general coefficient case of additive commutation always holds.
Known results
Tarski and Chang established partial cases, leaving the general coefficient case open.
August 2026 affirmative result
The paper “The Additive Arithmetic of Linear Orders” claims to prove the general coefficient case left open by Tarski and Chang, thereby completing the problem. The result is available as an arXiv preprint.
Current status (as of August 2026): The problem is resolved by the affirmative arXiv preprint, with no remaining mathematical case identified in the retrieved material.
Sources
Sources & referencesView supporting material
Primary source
Additional references
Solutions 0
Sign in to submit a solution.
No solutions have been posted yet.