  MV-monoids are algebras $\langle A,\vee,\wedge, \oplus,\odot, 0,1\rangle$
where $\langle A, \vee, \wedge, 0, 1\rangle$ is a bounded distributive lattice,
both $\langle A, \oplus, 0 \rangle$ and $\langle A, \odot, 1\rangle$ are
commutative monoids, and some further connecting axioms are satisfied. Every
MV-algebra in the signature $\{\oplus,\neg,0\}$ is term equivalent to an
algebra that has an MV-monoid as a reduct, by defining, as standard, $1:= \neg
0$, $x \odot y := \neg(\neg x \oplus\neg y)$, $x \vee y := (x \odot \neg y)
\oplus y$ and $x \wedge y := \neg(\neg x \vee \neg y)$. Particular examples of
MV-monoids are positive MV-algebras, i.e. the $\{\vee, \wedge, \oplus, \odot,
0, 1\}$-subreducts of MV-algebras. Positive MV-algebras form a peculiar
quasivariety in the sense that, albeit having a logical motivation (being the
quasivariety of subreducts of MV-algebras), it is not the equivalent
quasivariety semantics of any logic. In this paper, we study the lattices of
subvarieties of MV-monoids and of positive MV-algebras. In particular, we
characterize and axiomatize all almost minimal varieties of MV-monoids, we
characterize the finite subdirectly irreducible positive MV-algebras, and we
characterize and axiomatize all varieties of positive MV-algebras.
