A new mathematics preprint sets out an arithmetic for linear orders — abstract arrangements that can be combined by placing one order after another — and uses it to settle questions about when those combinations are the same. The results cover cancellation, division, commutativity and representation inside the class of all linear orders, known as LO.
The work is entirely formal. It does not analyze people, measurements or an empirical participant sample; its conclusions come from definitions and proofs about LO, ordered sums, embeddings and related algebraic structures.
Turning ordered sums into an arithmetic problem
The paper’s central tool is a generalized Euclidean algorithm, used for pairs of linear orders satisfying a one-sided weak commutativity relation. The algorithm supplies the information used in the paper’s division results.
The division theorem addresses a basic question: if finite repetitions of two orders match, what can be said about the orders themselves? The paper shows that, when the finite multiples are isomorphic, the two orders can be reduced to a common order after the coefficients are reduced by their greatest common divisor.
A related cancellation theorem says that a common positive finite multiple cannot hide a difference between the original orders. If two linear orders have the same such multiple, the paper concludes that the orders are isomorphic — mathematically, they have the same structure up to an order-preserving relabeling.
Answers to questions about when sums commute
Those tools feed into the paper’s treatment of Tarski’s Generalized Sum Problem. Within LO, the authors give an affirmative solution based on the information obtained from a run of the Euclidean algorithm.
The preprint also reports a general affirmative answer to Tarski’s positive-coefficient sum question. In its formal terms, if finite sums built from A and B are isomorphic for positive coefficients, then A+B is isomorphic to B+A.
The paper goes further by describing exactly when two orders commute under ordered sum. The criterion is expressed through embeddings — structure-preserving maps — between the right-infinite sums and the left-infinite sums. Either the required initial and final embeddings exist in one direction, or the corresponding conditions hold after exchanging A and B.
This gives the commuting question a structural answer rather than a numerical one. The issue is not simply how many copies of each order appear, but whether the infinite arrangements built from them fit into one another at their beginning and end in the required way.
From abstract orders to broader algebraic structures
Another part of the work studies representation: ways of realizing an abstract algebraic pattern inside LO. In a symmetric-divisor setting, the paper identifies alternatives involving absorption, common finite multiples, or a replacement representation based on translation along the real line by an irrational amount.
The preprint also gives a classification for commutative semigroups. It states that such a semigroup can be represented in (LO, +) exactly when it is isomorphic to a subgroup of an LO semigroup.
The authors additionally describe LO semigroups as naturally totally ordered under the ordering induced by their semigroup structure. That places an order relation around the algebra itself, alongside the order carried by each individual linear order.
The boundaries of the result
The strongest claims are limited to LO and the ordered-sum operation studied in the preprint. The current proof of Tarski’s Generalized Sum Problem does not establish the same result for arbitrary ordinal algebras; that extension is deferred to a forthcoming companion paper.
Some of the representation results also depend on a sufficiency hypothesis. The paper says that this condition cannot generally be removed, so a real-line representation should not be read as automatically sufficient in every non-terminating case.
These are theorem-level conclusions, not estimates drawn from statistical uncertainty or experimental validation. Their force depends on the formal assumptions and proofs used in the paper, including the stated conditions attached to particular representation results.
A preprint, not a finished consensus
The document is an arXiv preprint, version 1, dated 20 August 2026.
The paper reports support for its first author from the Momentum MSCA Programme, co-funded through the HORIZON-MSCA-2023-COFUND programme and the Secretariat of the Hungarian Academy of Sciences.
Taken together, the results present a formal arithmetic of LO: finite multiples can be canceled under the theorem’s conditions, certain sum questions have affirmative answers, commuting pairs have a structural criterion, and representable commutative semigroups receive an exact classification.
Paper data and sources
Original title: The Additive Arithmetic of Linear Orders
Authors: Garrett Ervin, Eric Paul
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text