PreprintCC BY 4.0arXiv
The AA problem for finite ordered sets
Abstract
For ordered sets X and Y, let YX denote the ordered set of order-preserving maps from X to Y, ordered pointwise. In 1984, D. Duffus asked whether AA≅ BB forces A≅ B when A and B are finite ordered sets. We prove that it does. The proof combines direct decompositions with an auxiliary ordering of connected ordered sets and a careful choice of intervals of maximum height. We then extend the reconstruction result to several forms of repeated exponentiation.
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BibTeX
@misc{farley2026problem,
title = {{The $A^A$ problem for finite ordered sets}},
author = {Jonathan David Farley and George Grätzer},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20817},
eprint = {2610.20817},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20817},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The A^A problem for finite ordered sets
AU - Jonathan David Farley
AU - George Grätzer
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.20817
UR - https://arxiv.org/abs/2610.20817
AB - For ordered sets X and Y, let Y^X denote the ordered set of order-preserving maps from X to Y, ordered pointwise. In 1984, D. Duffus asked whether A^A≅ B^B forces A≅ B when A and B are finite ordered sets. We prove that it does. The proof combines direct decompositions with an auxiliary ordering of connected ordered sets and a careful choice of intervals of maximum height. We then extend the reconstruction result to several forms of repeated exponentiation.
ER -