A quasivariety K of algebras has the joint embedding property (JEP) iff it is
generated by a single algebra A. It is structurally complete iff the free
countably generated algebra in K can serve as A. A consequence of this demand,
called "passive structural completeness" (PSC), is that the nontrivial members
of K all satisfy the same existential positive sentences. We prove that if K 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 K is
relatively semisimple then it is generated by one K-simple algebra. It is a
minimal quasivariety if, moreover, it is PSC but fails to unify some finite set
of equations. We also prove that a quasivariety of finite type, with a finite
nontrivial member, is PSC iff 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