4. Problems

Problem DD-1. How can we be sure that an ultraproduct of chains is a chain?

Problem DD-2. Prove the Corollary of § from Jónsson's Lemma.

Problem DD-3. Let F$_ 4$ be the Galois field of 4 elements, as a ring. Find all the SI members of Var$($F$_ 4)$, up to isomorphism. (You may use the fact that F$_ 4$ is congruence-distributive.)

Problem DD-4. True or false? ``Every lattice satisfies the same laws as its dual.'' If true, give a brief proof; if false, give a lattice that is a counterexample, with brief explanation. (Either way, it is not necessary to give any specific laws.)

Problem DD-5. Let ${\cal K}$ be the class of all lattices of width at most $5$. Show that each subdirectly irreducible member of Var$({\cal K})$ is in ${\cal K}$.