  The variety DMM of De Morgan monoids has just four minimal subvarieties. The
join-irreducible covers of these atoms in the subvariety lattice of DMM are
investigated. One of the two atoms consisting of idempotent algebras has no
such cover; the other has just one. The remaining two atoms lack nontrivial
idempotent members. They are generated, respectively, by 4-element De Morgan
monoids C4 and D4, where C4 is the only nontrivial 0-generated algebra onto
which finitely subdirectly irreducible De Morgan monoids may be mapped by
non-injective homomorphisms. The homomorphic pre-images of C4 within DMM
(together with the trivial De Morgan monoids) constitute a proper quasivariety,
which is shown to have a largest subvariety U. The covers of the variety V(C4)
within U are revealed here. There are just ten of them (all finitely
generated). In exactly six of these ten varieties, all nontrivial members have
C4 as a retract. In the varietal join of those six classes, every
subquasivariety is a variety-in fact, every finite subdirectly irreducible
algebra is projective. Beyond U, all covers of V(C4) [or of V(D4)] within DMM
are discriminator varieties. Of these, we identify infinitely many that are
finitely generated, and some that are not. We also prove that there are just 68
minimal quasivarieties of De Morgan monoids.
