Varieties of MV-monoids and positive MV-algebras

MV-monoids are algebras where is a bounded distributive lattice, both and are commutative monoids, and some further connecting axioms are satisfied. Every MV-algebra in the signature is term equivalent to an algebra that has an MV-monoid as a reduct, by defining, as standard, ≔ , ≔ , ≔ and ≔ . Particular examples of MV-monoids are positive MV-algebras, i.e., the -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.