Near-unanimity term

Remarks

Example: in the variety of lattices, $t(x, y, z) = (x \wedge y) \vee (y \wedge z) \vee (x \wedge z)$ is a near-unanimity term of arity $3$. Check: $t(y, x, x) = (y \wedge x) \vee x \vee (y \wedge x) = x$ by absorption, and the other two identities follow in the same way by symmetry of $t$. A near-unanimity term of arity $3$ is called a majority term. Non-example: the variety of groups has no near-unanimity term of any arity. The reason is that a variety with a near-unanimity term is congruence-distributive, while the group $\Z_2 \times \Z_2$ has a non-distributive lattice of normal subgroups. If $t$ is a near-unanimity term of arity $n$, then $s(x_1, \dots, x_{n + 1}) = t(x_1, \dots, x_n)$ is one of arity $n + 1$. The abbreviation NU term is common.