Catalog · 10.5281/zenodo.23458450
A countable scattered poset without a strongly maximal chain
- Martino Lupini · University of Bologna
18 October 2026 · CC BY 4.0 · latest of 2 versions
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A countable scattered poset without a strongly maximal chain
Martino Lupini
Abstract
A chain C in a partially ordered set is strongly maximal if |D \ C| ≤ |C \ D| for every chain D. We construct a countably infinite poset with no infinite antichain and no strongly maximal chain. The poset is scattered: none of its chains contains a copy of the rational order. This gives a negative answer, already in the countable scattered case, to the strong-chain question posed by Hollom in 2024 and restated in 2026. The construction replaces each point (x, y, n) of Hollom’s layered poset by a finite chain of length 2^x + 2^y. A finite path exchange restricts the possible chains, after which a further finite exchange always improves them. We also analyze a width-two example of Hollom showing that the chain-extension inclusion used in an earlier claimed positive proof need not hold.
GitHub repository: https://github.com/eulogica/strongly-maximal-chains
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BibTeX
@unpublished{Lupini202623458450,
title = {A countable scattered poset without a strongly maximal chain},
author = {Martino Lupini},
year = {2026},
doi = {10.5281/zenodo.23458450},
note = {Zenodo deposit, latest of 2}
}