Varieties of MV-monoids and positive MV-algebras

Abstract

MV-monoids are algebras $〈A,\vee ,\wedge ,\oplus ,\odot ,0,1〉$ where $〈A,\vee ,\wedge ,0,1〉$ is a bounded distributive lattice, both $〈A,\oplus ,0〉$ and $〈A,\odot ,1〉$ 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.