User:SnEk/Jónsson terms

Jónsson terms are terms that describes congruence distributive varieties. They have been discovered by Bjarni Jónsson in 1967.

Theorem. The following conditions are equivalent for a~variety of algebras $$\mathcal V$$:


 * 1) every algebra in $$\mathcal V$$ has distributive congruence lattice,
 * 2) the 3-generated free algebra in $$\mathcal V$$ has distibutive congruence lattice,
 * 3) there exists $$n\in \mathbb N$$, and ternary terms $$p_0, \dots, p_n$$ such that

$$p_0(x,y,z) = x, p_n(x,y,z) = z,$$

$$p_i(x,y,x) = x,$$ for all $$i$$,

$$p_i(x,x,y) = p_{i+1}(x,x,y)$$, for $$i$$ even,

$$p_i(x,y,y) = p_{i+1}(x,y,y)$$, for $$i$$ odd.