Varieties of MV-monoids and positive MV-algebras

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.