1. Mal'tsev conditions

Other theorems of the same general kind have been discovered, where a condition on a variety is characterized using the existence of terms obeying laws of some sort. Such a condition is now called a ``Mal'tsev condition''. Some typical examples, in additional to congruence permutability, are

• $V$ is congruence-distributive. In other words, for any ${\cal A} \in V$, Con$(A)$ is a distributive lattice.

  Example: The variety of all lattices; the variety of all Boolean algebras.

• $V$ is congruence-modular. In other words, for any ${\cal A} \in V$, Con$(A)$ is a modular lattice.

  Since the distributive law implies the modular law, any congruence-distributive variety is also congruence-modular. Also, we have:

  Proposition. Any congruence-permutable variety is congruence-modular.

• $V$ is arithmetic (``arithmet$'$ic''). This means that $V$ is both congruence-permutable and congruence-distributive.

  Example: The variety of rings generated by a finite field.