The Additive Arithmetic of Linear Orders
Abstract
We present a systematic development of the arithmetic of the class of linear orders under the ordered sum (LO, +) and prove a number of new results. Our approach is based on a Euclidean algorithm for pairs of linear orders that almost additively commute. Among our results: (i.) We generalize and give unified proofs of the main classical theorems for (LO, +), including Lindenbaum's division theorem for (LO, +) and a representation theorem for additively commuting pairs of linear orders due to Aronszajn. (ii.) We solve the following problem, posed by Tarski in 1956: is it true that for every pair of linear orders $A, B$ and quadruple of natural numbers n, m, k, l ≥ 1, if nA + mB ≅ kB + lA then A + B ≅ B + A? Tarski and Chang showed the answer is yes for certain choices of the coefficients $n, m, k, l$. We show the answer is yes in general. (iii.) We prove the following characterization of the additively commuting pairs in LO: A + B ≅ B + A if and only if ωA embeds initially in ωB and ω* A embeds finally in ω* B, or vice versa. We show this can be viewed as a correctly revised version of a refuted conjecture of Tarski. (iv.) We characterize the commutative semigroups (S, ⊕) that can be represented in (LO, +) and show in particular they are all subsemigroups of naturally totally ordered semigroups in the sense of Clifford.
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BibTeX
@misc{ervin2026additive,
title = {{The Additive Arithmetic of Linear Orders}},
author = {Garrett Ervin and Eric Paul},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.20309},
eprint = {2608.20309},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.20309},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - The Additive Arithmetic of Linear Orders
AU - Garrett Ervin
AU - Eric Paul
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2608.20309
UR - https://arxiv.org/abs/2608.20309
AB - We present a systematic development of the arithmetic of the class of linear orders under the ordered sum (LO, +) and prove a number of new results. Our approach is based on a Euclidean algorithm for pairs of linear orders that almost additively commute. Among our results: (i.) We generalize and give unified proofs of the main classical theorems for (LO, +), including Lindenbaum's division theorem for (LO, +) and a representation theorem for additively commuting pairs of linear orders due to Aronszajn. (ii.) We solve the following problem, posed by Tarski in 1956: is it true that for every pair of linear orders $A, B$ and quadruple of natural numbers n, m, k, l ≥ 1, if nA + mB ≅ kB + lA then A + B ≅ B + A? Tarski and Chang showed the answer is yes for certain choices of the coefficients $n, m, k, l$. We show the answer is yes in general. (iii.) We prove the following characterization of the additively commuting pairs in LO: A + B ≅ B + A if and only if ωA embeds initially in ωB and ω^* A embeds finally in ω^* B, or vice versa. We show this can be viewed as a correctly revised version of a refuted conjecture of Tarski. (iv.) We characterize the commutative semigroups (S, ⊕) that can be represented in (LO, +) and show in particular they are all subsemigroups of naturally totally ordered semigroups in the sense of Clifford.
ER -