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The Additive Arithmetic of Linear Orders — arXiv:2608.20309

Source: arXiv (en)
Original: https://arxiv.org/abs/2608.20309
License (as archived): CC0-1.0
Archived: 2026-10-18T12:02:01.646Z
SHA-256 of the archived text: a422a7d949e0d8d233f3be1920a97af6c394a620345aabbb1b5502cfcfbfe3df
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How to cite

The Additive Arithmetic of Linear Orders — arXiv:2608.20309 — arXiv (en), https://arxiv.org/abs/2608.20309 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_03bcd9ec366e (Metraipolis archive dc_03bcd9ec366e, sha256 a422a7d949e0d8d233f3be1920a97af6c394a620345aabbb1b5502cfcfbfe3df)

BibTeX

@misc{dc03bcd9ec366e,
  title = {The Additive Arithmetic of Linear Orders — arXiv:2608.20309},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2608.20309},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_03bcd9ec366e; sha256 a422a7d949e0d8d233f3be1920a97af6c394a620345aabbb1b5502cfcfbfe3df},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2608.20309. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

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 $n, m, k, l \geq 1$, if $nA + mB \cong kB + lA$ then $A + B \cong B + A$? 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 $LO$: $A + B \cong B + A$ if and only if $\omega A$ embeds initially in $\omega B$ and $\omega^* A$ embeds finally in $\omega^* B$, 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 $(S, \oplus)$ that can be represented in $(LO, +)$ and show in particular they are all subsemigroups of naturally totally ordered semigroups in the sense of Clifford.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-08-21 in math.LO).
Categorie arXiv: Logic (math.LO).
Pagina del lavoro: https://arxiv.org/abs/2608.20309 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-18T12:00:23.714Z dalla pagina https://arxiv.org/catchup/math.LO/2026-08-21?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).