The Dimension of Divisibility Orders and Multiset Posets
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11083_2023_Article_9653.pdf
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Author(s)
Haiman, Milan
Date Issued
November 22, 2023
Publisher
Springer Netherlands
Citation
Haiman, Milan. 2023. "The Dimension of Divisibility Orders and Multiset Posets."
Version
Final published version
Abstract
Abstract
The Dushnik–Miller dimension of a poset P is the least d for which P can be embedded into a product of d chains. Lewis and Souza isibility order on the interval of integers
$$[N/\kappa , N]$$
[
N
/
κ
,
N
]
is bounded above by
$$\kappa (\log \kappa )^{1+o(1)}$$
κ
(
log
κ
)
1
+
o
(
1
)
and below by
$$\Omega ((\log \kappa /\log \log \kappa )^2)$$
Ω
(
(
log
κ
/
log
log
κ
)
2
)
. We improve the upper bound to
$$O((\log \kappa )^3/(\log \log \kappa )^2).$$
O
(
(
log
κ
)
3
/
(
log
log
κ
)
2
)
.
We deduce this bound from a more general result on posets of multisets ordered by inclusion. We also consider other divisibility orders and give a bound for polynomials ordered by divisibility.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1007/s11083-023-09653-7
