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.

References

Maltsev condition