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

TL;DR

This paper investigates the conditions under which almost structurally complete quasivarieties are actually structurally complete, providing a general solution and characterizations specifically for discriminator varieties, with implications for certain logical systems.

Contribution

It offers a general method to determine when almost structurally complete quasivarieties are structurally complete and characterizes structurally complete discriminator varieties.

Findings

A general solution to the structural completeness problem in quasivarieties.

Characterization of structurally complete discriminator varieties.

Logical consequence: certain propositional logics are maximal iff structurally complete.

Abstract

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.