0. Mal'tsev's Theorem

Two congruences $\alpha, \beta$ on an algebra $A$ are said to permute if they commute: $\alpha \beta = \beta \alpha$.

Here $\alpha \beta$ is the composition, given by $x \; \alpha \beta \; z$ if and only if there exists $y$ with $x \;\alpha \;y \;\beta \;z$. Notice that $\alpha \beta$ and $\beta \alpha$ are not necessarily equivalence relations.

A variety $V$ is said to be congruence permutable if in any algebra in $V$ any two congruence relations permute. The famous Russian algebraist Mal'tsev (also transliterated as Mal'cev) gave this characterization:

0.1 Theorem (Mal'tsev) For a variety $V$, the following are equivalent:

(a) $V$ is congruence-permutable;

(b) there is a term $p(x,y,z)$ such that in $V$ these laws hold:

$p(x,x,z) = z$,

$p(x,z,z) = x$.

For example, the variety of groups satisfies this condition with $p(x,y,z) = x y ^ {-1} z$. The same term, written additively as $x - y + z$, then works for rings, since rings have an additive group.