PreprintCC BY 4.0arXiv
On canonicity of almost linear minimal orders
Abstract
We prove that if a minimal ordered structure (M, <, …) with infinite chains in an arbitrary language extending the language of strict orders interprets (in some power Mn) the linear order (ω, <), then this fact can be witnessed for n=1 via the incomparability relation of an <-definable strict order R; that R is unique up to ``almost equality'', i.e., finite rearrangements of elements of M; and that R can be defined from < in a constructive way. We also show that several variants of the question whether a minimal ordered structure interprets an infinite linear order are all equivalent.
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BibTeX
@misc{jagiella2026canonicity,
title = {{On canonicity of almost linear minimal orders}},
author = {Grzegorz Jagiella},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.36313},
eprint = {2609.36313},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.36313},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On canonicity of almost linear minimal orders
AU - Grzegorz Jagiella
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.36313
UR - https://arxiv.org/abs/2609.36313
AB - We prove that if a minimal ordered structure (M, <, …) with infinite chains in an arbitrary language extending the language of strict orders interprets (in some power M^n) the linear order (ω, <), then this fact can be witnessed for n=1 via the incomparability relation of an <-definable strict order R; that R is unique up to ``almost equality'', i.e., finite rearrangements of elements of M; and that R can be defined from < in a constructive way. We also show that several variants of the question whether a minimal ordered structure interprets an infinite linear order are all equivalent.
ER -