Abstract:
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 nitely 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 nitely 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 nite 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 in nitely
many that are nitely generated, and some that are not. We also prove
that there are just 68 minimal quasivarieties of De Morgan monoids.
