PreprintCC BY-SA 4.0arXiv
Grün's Lemma for Semiassociative Mal'cev Algebras
Abstract
We extend recent work on analogues of Grün's Lemma from skew-braces to all semiassociative Mal'cev algebras which is a broad class of algebras encompassing all expansions of groups. The direct analogue of Grün's Lemma is known to fail for skew-braces; thus, the more general result is a characterization for perfect algebras of when the upper-central series stabilizes at the second-center. The proof is an application of the description of generating sets for commutators in Mal'cev algebras.
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BibTeX
@misc{wires2026grüns,
title = {{Grün's Lemma for Semiassociative Mal'cev Algebras}},
author = {Alexander Wires},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35396},
eprint = {2609.35396},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35396},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Grün's Lemma for Semiassociative Mal'cev Algebras
AU - Alexander Wires
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35396
UR - https://arxiv.org/abs/2609.35396
AB - We extend recent work on analogues of Grün's Lemma from skew-braces to all semiassociative Mal'cev algebras which is a broad class of algebras encompassing all expansions of groups. The direct analogue of Grün's Lemma is known to fail for skew-braces; thus, the more general result is a characterization for perfect algebras of when the upper-central series stabilizes at the second-center. The proof is an application of the description of generating sets for commutators in Mal'cev algebras.
ER -