Mal'cev condition

Remarks

Example: the existence of a Mal'cev term is a strong Mal'cev condition, with one ternary symbol $p$ and $\Sigma = \set{p(x, y, y) = x, \ p(x, x, y) = y}$. The variety of groups satisfies it with $t(x, y, z) = x y^{-1} z$, since $x y^{-1} y = x$ and $x x^{-1} z = z$. The existence of Jónsson terms for some $n$ is a Mal'cev condition, with one strong Mal'cev condition $U_n$ for each $n$. Non-example: local finiteness of a variety is not Mal'cev definable. The variety of sets, with no operations, is locally finite, and its terms are the projections. If the projections satisfy $\Sigma$, then the projections of every variety satisfy $\Sigma$, so every variety would satisfy the condition. But the variety of groups is not locally finite.