Extension properties for partial permutations
Abstract
Motivated by the EPPA problem for finite tournaments,we consider various extension properties for partial permutations. We show that for any set Π of prime numbers, the Π-extension property is equivalent to the Π-LERF. As a consequence of known results, the odd-extension property is then equivalent to the EPPA for finite tournaments. To study the Π-extension property, we reformate the property using the concepts of EP problems and Π-solutions. We then show that the existence of a Π-solution for an EP problem depends entirely on its fundamental group. In particular, when the fundamental group is trivial or cyclic, the EP problem has a Π-solution for any Π. These extend some known results of Huang, Pawliuk, Sabok and Wise [HPSW19]. We give examples of EP problems without nilpotent-solutions; they witness that the p-extension property and the p-LERF fail for any prime p. Then we consider a special kind of EP problems whose fundamental groups have two generators. We show that any 1-weakly wandering problem has a Π-solution for any Π. For 2-weakly wandering problems, we show that they all have odd-solutions and we completely characterize those without nilpotent-solutions.
Related
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BibTeX
@misc{etedadialiabad2026extension,
title = {{Extension properties for partial permutations}},
author = {Mahmood Etedadialiabad and Su Gao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04870},
eprint = {2610.04870},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04870},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Extension properties for partial permutations
AU - Mahmood Etedadialiabad
AU - Su Gao
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.04870
UR - https://arxiv.org/abs/2610.04870
AB - Motivated by the EPPA problem for finite tournaments,we consider various extension properties for partial permutations. We show that for any set Π of prime numbers, the Π-extension property is equivalent to the Π-LERF. As a consequence of known results, the odd-extension property is then equivalent to the EPPA for finite tournaments. To study the Π-extension property, we reformate the property using the concepts of EP problems and Π-solutions. We then show that the existence of a Π-solution for an EP problem depends entirely on its fundamental group. In particular, when the fundamental group is trivial or cyclic, the EP problem has a Π-solution for any Π. These extend some known results of Huang, Pawliuk, Sabok and Wise [HPSW19]. We give examples of EP problems without nilpotent-solutions; they witness that the p-extension property and the p-LERF fail for any prime p. Then we consider a special kind of EP problems whose fundamental groups have two generators. We show that any 1-weakly wandering problem has a Π-solution for any Π. For 2-weakly wandering problems, we show that they all have odd-solutions and we completely characterize those without nilpotent-solutions.
ER -