Varieties of MV-monoids and positive MV-algebras

In this paper we investigate MV-monoids and their subquasivarieties. MV-monoids are algebras A,,, ⊕,, 0,1 where A, , , 0, 1 is a bounded distributive lattice, A, ⊕, 0 and A, , 1 are commutative monoids, and some further connecting axioms are satisfied. Every MV-algebra in the signature \⊕,,0\ is term equivalent to an algebra that has an MV-monoid as a reduct, by defining, as standard, 1:= 0, x y := ( x ⊕ y), x y := (x y) ⊕ y and x y := ( x y). Particular examples of MV-monoids are positive MV-algebras, i.e. the \, , ⊕, , 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 lattice of subvarieties of MV-monoids and describe the lattice of subvarieties of positive MV-algebras. We characterize the finite subdirectly irreducible positive MV-algebras. Furthermore, we axiomatize all varieties of positive MV-algebras.