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Given an algebra $\mathbf{A}$, a pair of congruences $ \alpha ,\beta \in Con(\mathbf {A})$ are said to permute when $ \alpha \circ \beta =\beta \circ \alpha$, and an algebra $\mathbf{A}$ is called ...

By a theorem of Hagemann and Mitschke, a condition (A) that a variety $\mathcal{V}$ is congruence $n$-permutable, is equivalent to a condition (B) that there exist ternary terms $p_1,\dots,p_{n-1}$ ...