Varieties of MV-monoids and positive MV-algebras
(2025) Journal of Algebra — Vol. 677, p. 690-744 (2025)
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- Abstract
- MV-monoids are algebras 〈A,∨,∧,⊕,⊙,0,1〉 where 〈A,∨,∧,0,1〉 is a bounded distributive lattice, both 〈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 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.
- Affiliations
University of Birmingham Università di Siena Charles University
Citations
Abbadini, M., Aglianò, P., & Fioravanti, S. (2025). Varieties of MV-monoids and positive MV-algebras. Journal of Algebra, 677, 690-744. https://doi.org/10.1016/j.jalgebra.2025.04.027 (Original work published 2025)