3. Problems

Problem CC-1. Prove Mal'tsev's theorem.

Problem CC-2. Prove Pixley's theorem.

Problem CC-3. Another Mal'tsev condition:

(a) Show that the following are equivalent for a variety $V$:

(i) $V$ has a majority term;

(ii) intersections of congruences distribute over composition:

$\alpha \cap (\beta \gamma) = (\alpha \cap \beta)(\alpha \cap \gamma)$.

(b) Show that a variety with a majority term is congruence-distributive (the case $n=2$ of Jónsson's theorem). (Method: Use (ii), generalized to compositions of more than two congruences by an easy induction. Recall that $\alpha \vee \beta$ is the union of $\alpha \beta$, $\alpha \beta \alpha$, $\alpha \beta \alpha \beta$, etc.)