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PreprintCC BY 4.0arXiv

Does the Endomorphism Ordered Set of a Finite Ordered Set Determine the Ordered Set? The Cases of Height $1$ and "Trebled" Ordered Sets

  • Jonathan David Farley
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Abstract

For ordered sets X and Y, let YX denote the ordered set of order-preserving maps from X to Y, where f≤ g in YX if f(x)≤ g(x) for all x∈ X. Let P and Q be finite ordered sets such that PP≅ QQ. It is proven that P≅ Q if P or Q has height at most $1$ or if P and Q are ordered sets of the following form: replace each element of an ordered set with a three-element antichain. The latter is an elaboration of a proof of Tim Campion.

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Cite

BibTeXRIS
BibTeX
@misc{farley2026does,
  title = {{Does the Endomorphism Ordered Set of a Finite Ordered Set Determine the Ordered Set? The Cases of Height $1$ and "Trebled" Ordered Sets}},
  author = {Jonathan David Farley},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.38532},
  eprint = {2609.38532},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.38532},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Does the Endomorphism Ordered Set of a Finite Ordered Set Determine the Ordered Set? The Cases of Height $1$ and "Trebled" Ordered Sets
AU  - Jonathan David Farley
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.38532
UR  - https://arxiv.org/abs/2609.38532
AB  - For ordered sets X and Y, let Y^X denote the ordered set of order-preserving maps from X to Y, where f≤ g in Y^X if f(x)≤ g(x) for all x∈ X. Let P and Q be finite ordered sets such that P^P≅ Q^Q. It is proven that P≅ Q if P or Q has height at most $1$ or if P and Q are ordered sets of the following form: replace each element of an ordered set with a three-element antichain. The latter is an elaboration of a proof of Tim Campion.
ER  -