2024/05/14 by Marco Abbadini, Abbadini, Marco, Paolo Aglianò +3
Computer Science · Decision Sciences · #03C05 #06D35 #06F05 #08B15 #08B26 #08C15 #Advanced Algebra and Logic #FOS: Mathematics #Fuzzy and Soft Set Theory #Logic (math.LO) #Rings and Algebras (math.RA) #Rough Sets and Fuzzy Logic
paper · pdf · doi:10.48550/arxiv.2405.08471
openalex publication_date 2024/05/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
MV-monoids are algebras ⟨ A,\vee,\wedge, ⊕,\odot, 0,1⟩ where ⟨ A, \vee, \wedge, 0, 1⟩ is a bounded distributive lattice, both ⟨ A, ⊕, 0 ⟩ and ⟨ A, \odot, 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 \odot y := ¬(¬ x ⊕¬ y), x \vee y := (x \odot ¬ y) ⊕ y and x \wedge y := ¬(¬ x \vee ¬ y). Particular examples of MV-monoids are positive MV-algebras, i.e. the \\vee, \wedge, ⊕, \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.