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.

对于一个变种\（{\ mathcal {V}} \\），最近发现二进制产品在本地与任意的均衡器进行换向，即在点\（\ pi：\ mathrm {Pt}（ {\ mathbb {C}}）\ rightarrow {\ mathbb {C}} \），当且仅当Gumm的移位引理在\（{\ mathcal {V}} \）中保持回调。在本文中，我们建立了将通用代数中的所谓三角引理与一定的分类反对换性条件联系起来的相似结果。特别是，我们证明了这种反对易性及其局部形式是Mal'tsev条件，局部形式等效于回撤时的三角引理。因此，每个局部反交换变种\（{\ mathcal {V}} \）在Duda的意义上具有可直接分解的同余类，如果\（{\ mathcal {V}} \\）是幂等的，则反之成立。