We use an algebraic argument to prove that there are exactly two premaximal extensions of \’s consequence. We also show that one of these extensions is the minimal structurally complete extension of the unique maximal paraconsistent extension of \. Precisely, we show that there are exactly two covers of the variety of Boolean algebras in the lattice of quasivarieties of Sugihara algebras and that there is a unique minimal paraconsistent quasivariety in that lattice. We also obtain a corollary stating that (...) the set of paraconsistent extensions of \ forms a complete sublattice of the lattice of all \’s extensions. (shrink)

It is proved that the relevance logic \ has no structurally complete consistent axiomatic extension, except for classical propositional logic. In fact, no other such extension is even passively structurally complete.

Hereditary structural completeness is established for a range of substructural logics, mainly without the weakening rule, including fragments of various relevant or many-valued logics. Also, structural completeness is disproved for a range of systems, settling some previously open questions.