Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Varieties of MV-monoids and positive MV-algebras

Marco Abbadini, Paolo Aglianò, Stefano Fioravanti

TL;DR

This work develops a comprehensive algebraic account of two negation-free variants of MV-algebras—MV-monoids and positive MV-algebras—by leveraging Mundici's Γ-equivalence between unital ell-monoids and MV-monoids. It delivers a complete lattice-theoretic picture: a precise description of the bottom and almost minimal varieties of MV-monoids, a finite characterization of subdirectly irreducible positive MV-algebras (with a distinction between absolute and relative irreducibility for C^Δ and C^∇), and a full axiomatization of all varieties of positive MV-algebras. The positive MV-algebras are shown to be exactly the varieties generated by finitely many reducts of finite nontrivial MV-chains, with a bijection to divisor-closed finite sets of natural numbers, enabling explicit finite bases of equations. The results bridge categorical, lattice-theoretic, and equational approaches, offering a concrete framework for further exploration of algebraic and logical aspects of these many-valued structures.

Abstract

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

Varieties of MV-monoids and positive MV-algebras

TL;DR

This work develops a comprehensive algebraic account of two negation-free variants of MV-algebras—MV-monoids and positive MV-algebras—by leveraging Mundici's Γ-equivalence between unital ell-monoids and MV-monoids. It delivers a complete lattice-theoretic picture: a precise description of the bottom and almost minimal varieties of MV-monoids, a finite characterization of subdirectly irreducible positive MV-algebras (with a distinction between absolute and relative irreducibility for C^Δ and C^∇), and a full axiomatization of all varieties of positive MV-algebras. The positive MV-algebras are shown to be exactly the varieties generated by finitely many reducts of finite nontrivial MV-chains, with a bijection to divisor-closed finite sets of natural numbers, enabling explicit finite bases of equations. The results bridge categorical, lattice-theoretic, and equational approaches, offering a concrete framework for further exploration of algebraic and logical aspects of these many-valued structures.

Abstract

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.
Paper Structure (9 sections, 68 theorems, 85 equations, 9 figures)

This paper contains 9 sections, 68 theorems, 85 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Preliminaries: lattice-ordered monoids, MV-monoids and positive MV-algebras
  3. Subdirectly irreducible MV-monoids
  4. The almost minimal varieties of MV-monoids
  5. Above the almost minimal varieties of MV-monoids
  6. Subdirectly irreducible positive MV-algebras
  7. Varieties of positive MV-algebras
  8. Axiomatizations
  9. Conclusions

Key Result

Theorem 2.9

The functor $\Gamma \colon \mathsf{u\ell M} \to \mathsf{MVM}$ is an equivalence of categories.

Figures (9)

  • Figure 1: The bottom part of $\Lambda(\mathsf{MVM})$
  • Figure 2: A portion of $\Lambda(\mathsf{MVM})$
  • Figure 3: the 3-element MV-monoids
  • Figure 4: the 4-element MV-monoids satisfying the conclusion of \ref{['subdMVM']}
  • Figure 5: $\operatorname{Con}({\textbf{\upshape L}}_3)$
  • ...and 4 more figures

Theorems & Definitions (150)

  • Definition 2.1
  • Example 2.2
  • Remark 2.3
  • Definition 2.4
  • Example 2.5
  • Remark 2.6
  • Remark 2.7
  • Definition 2.8
  • Theorem 2.9: Abbadini2021b
  • Definition 2.10: Abbadinietal2022
  • ...and 140 more