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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 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.

On structural completeness versus almost structural completeness problem: A discriminator varieties case study