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Generalizing the AA Problem for Finite Ordered Sets
Abstract
Let X and Y be finite ordered sets and let XY denote the ordered set of order-preserving maps from Y to X. Let t be a~term formed from a~single variable by exponentiation. We prove that, for every term t with at most twelve variable occurrences and arbitrary finite ordered sets A,B, t(A)≅ t(B) implies that A≅ B. General constructions and reconstruction principles are developed first; the second part gives the occurrence-by-occurrence proofs. At twelve occurrences the proof covers 4,766 interchange normal forms, representing all 58,786 binary parenthesizations, including 40 exceptional forms. Two additional unbounded reconstruction families, complete finite audit data, and 457 exact height-comparison certificates are included.
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BibTeX
@misc{gratzer2026generalizing,
title = {{Generalizing the $A^A$ Problem for Finite Ordered Sets}},
author = {George Grätzer},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03986},
eprint = {2610.03986},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03986},
}
RIS
TY - UNPB
TI - Generalizing the A^A Problem for Finite Ordered Sets
AU - George Grätzer
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03986
UR - https://arxiv.org/abs/2610.03986
AB - Let X and Y be finite ordered sets and let X^Y denote the ordered set of order-preserving maps from Y to X. Let t be a~term formed from a~single variable by exponentiation. We prove that, for every term t with at most twelve variable occurrences and arbitrary finite ordered sets A,B, t(A)≅ t(B) implies that A≅ B. General constructions and reconstruction principles are developed first; the second part gives the occurrence-by-occurrence proofs. At twelve occurrences the proof covers 4,766 interchange normal forms, representing all 58,786 binary parenthesizations, including 40 exceptional forms. Two additional unbounded reconstruction families, complete finite audit data, and 457 exact height-comparison certificates are included.
ER -