For which pairs of linear orders $A$ and $B$ are the sums $A + B$ and $B + A$ isomorphic? In the 1930s, Tarski conjectured that $A + B = B + A$ if and only if

(i.) There is an order $C$ and natural numbers $n$ and $m$ such that $A = nC$ and $B = mC$, or

(ii.) There is an order $M$ such that $B = \omega A + M + \omega^* A$, or  

(iii.) There is an order $N$ such that $A = \omega B + N + \omega^* B$. 

Notably, these conditions on $A$ and $B$ are “arithmetic” in the sense that they are expressed in terms of finitary and $\omega$-ary sums of linear orders. 

Tarski proved his conjecture over the class of scattered linear orders, but Lindenbaum was able to produce a non-scattered counterexample. Building on Lindenbaum’s work, Aronszajn found a structural characterization of all additively commuting pairs of linear orders. 

Aronszajn’s characterization is somewhat complicated: in modern language it can be described in terms of orbit equivalence relations of groups of translations on $\mathbb{R}$. Tarski lamented that Aronszajn’s result could not be formulated arithmetically — that is, purely in terms of sums — and the line of work was abandoned. 

Building on our recent work on sums of linear orders, Eric Paul and I showed that there is an arithmetic condition equivalent to commutativity for linear orders. And in fact, the condition is a natural extension of the one appearing in Tarski’s original conjecture. In this talk, I will state our result, outline the proof, and discuss some related problems.