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The Malt'sev condition for groups is not categorical
Abstract
In this paper we prove that the Mal'tsev condition expressing the existence of group operations is not equivalent to a categorical property. More precisely, using pseudo-invertible idempotents, we construct two varieties 𝒱 and 𝒱(f) which are equivalent as categories, and such that 𝒱 admits group operations, whereas 𝒱(f) does not. We also show that those Mal'tsev conditions which are equivalent to a categorical property have a trivial spectrum.
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BibTeX
@misc{stewart2026maltsev,
title = {{The Malt'sev condition for groups is not categorical}},
author = {Carlyle Stewart},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32399},
eprint = {2610.32399},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32399},
}
RIS
TY - UNPB
TI - The Malt'sev condition for groups is not categorical
AU - Carlyle Stewart
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32399
UR - https://arxiv.org/abs/2610.32399
AB - In this paper we prove that the Mal'tsev condition expressing the existence of group operations is not equivalent to a categorical property. More precisely, using pseudo-invertible idempotents, we construct two varieties 𝒱 and 𝒱(f) which are equivalent as categories, and such that 𝒱 admits group operations, whereas 𝒱(f) does not. We also show that those Mal'tsev conditions which are equivalent to a categorical property have a trivial spectrum.
ER -