Abstract
For a variety \({\mathcal {V}}\), it has been recently shown that binary products commute with arbitrary coequalizers locally, i.e., in every fibre of the fibration of points \(\pi : \mathrm {Pt}({\mathbb {C}}) \rightarrow {\mathbb {C}}\), if and only if Gumm’s shifting lemma holds on pullbacks in \({\mathcal {V}}\). In this paper, we establish a similar result connecting the so-called triangular lemma in universal algebra with a certain categorical anticommutativity condition. In particular, we show that this anticommutativity and its local version are Mal’tsev conditions, the local version being equivalent to the triangular lemma on pullbacks. As a corollary, every locally anticommutative variety \({\mathcal {V}}\) has directly decomposable congruence classes in the sense of Duda, and the converse holds if \({\mathcal {V}}\) is idempotent.
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Hoefnagel, M. Anticommutativity and the triangular lemma. Algebra Univers. 82, 19 (2021). https://doi.org/10.1007/s00012-021-00710-z
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DOI: https://doi.org/10.1007/s00012-021-00710-z
Keywords
- Anticommutativity
- Triangular lemma
- Shifting lemma
- Congruence distributivity
- Majority categories
- Directly decomposable congruence classes