  We study the following problem: Determine which almost structurally complete
quasivarieties are structurally complete. We propose a general solution to this
problem and then a solution in the semisimple case. As a consequence, we obtain
a characterization of structurally complete discriminator varieties.
  An interesting corollary in logic follows: Let $L$ be a consistent
propositional logic/deductive system in the language with formulas for verum,
which is a theorem, and falsum, which is not a theorem. Assume also that $L$
has an adequate semantics given by a discriminator variety. Then $L$ is
structurally complete if and only if it is maximal. All such logics/deductive
systems are almost structurally complete.
