Singly generated quasivarieties and residuated structures

A quasivariety (Formula presented.) of algebras has the joint embedding property (JEP) if and only if it is generated by a single algebra A. It is structurally complete if and only if the free ℵ0-generated algebra in (Formula presented.) can serve as A. A consequence of this demand, called ‘passive structural completeness’ (PSC), is that the nontrivial members of (Formula presented.) all satisfy the same existential positive sentences. We prove that if (Formula presented.) is PSC then it still has the JEP, and if it has the JEP and its nontrivial members lack trivial subalgebras, then its relatively simple members all belong to the universal class generated by one of them. Under these conditions, if (Formula presented.) is relatively semisimple then it is generated by one (Formula presented.) -simple algebra. We also prove that a quasivariety of finite type, with a finite nontrivial member, is PSC if and only if its nontrivial members have a common retract. The theory is then applied to the variety of De Morgan monoids, where we isolate the sub(quasi)varieties that are PSC and those that have the JEP, while throwing fresh light on those that are structurally complete. The results illuminate the extension lattices of intuitionistic and relevance logics.