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Subvarieties of pointed Abelian l-groups

Filip Jankovec

Abstract

This paper provides a complete classification of the subvarieties and subquasivarieties of pointed Abelian lattice-ordered groups ($\ell$-groups) that are generated by their totally ordered members. We present two complementary approaches to achieve this classification. First, using purely $\ell$-group-theoretic methods, we analyze the structure of lexicographic products and values to identify all join-irreducible members of the lattice of subvarieties of positively pointed Abelian $\ell$-groups. We provide a novel equational basis for each of these subvarieties, leading to a complete description of the entire subvariety lattice. As a direct application, our $\ell$-group-theoretic classification yields an alternative, self-contained proof of Komori's classification of subvarieties of MV-algebras. Second, we explore the connection to MV-algebras via Mundici's $Γ$ functor. We prove that this functor preserves universal classes, a result of independent model-theoretic interest. This allows us to lift the classification of universal classes of totally ordered MV-algebras, due to Gispert, to a complete classification of universal classes of totally ordered pointed Abelian $\ell$-groups. As a direct consequence, we obtain a full description of the corresponding lattice of subquasivarieties.

Subvarieties of pointed Abelian l-groups

Abstract

This paper provides a complete classification of the subvarieties and subquasivarieties of pointed Abelian lattice-ordered groups (-groups) that are generated by their totally ordered members. We present two complementary approaches to achieve this classification. First, using purely -group-theoretic methods, we analyze the structure of lexicographic products and values to identify all join-irreducible members of the lattice of subvarieties of positively pointed Abelian -groups. We provide a novel equational basis for each of these subvarieties, leading to a complete description of the entire subvariety lattice. As a direct application, our -group-theoretic classification yields an alternative, self-contained proof of Komori's classification of subvarieties of MV-algebras. Second, we explore the connection to MV-algebras via Mundici's functor. We prove that this functor preserves universal classes, a result of independent model-theoretic interest. This allows us to lift the classification of universal classes of totally ordered MV-algebras, due to Gispert, to a complete classification of universal classes of totally ordered pointed Abelian -groups. As a direct consequence, we obtain a full description of the corresponding lattice of subquasivarieties.

Paper Structure

This paper contains 8 sections, 44 theorems, 40 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The lexicographic decompositions
  4. Characterization of varieties generated by a single finitely generated totally ordered pointed Abelian l-group
  5. Axiomatization of subvarieties of pAL
  6. Applying the Mundici functor
  7. Gispert's classification of universal classes of totally ordered MV-algebras
  8. Future Work

Key Result

Theorem 2.1

Let ${{\mathbb K}}$ be a class of algebras of the same signature. The following conditions are equivalent: Moreover, if algebras in ${{\mathbb K}}$ have a group reduct, we can add the following: Furthermore, if ${{\mathbb K}}$ satisfies any of the equivalent conditions above (i.e., if ${{\mathbb K}}$ is a variety), then:

Figures (1)

  • Figure 1: The lattice of join irreducible subvarieties of $p\mathbb{AL}$.

Theorems & Definitions (78)

  • Theorem 2.1: Birkhoff Birkhoff:AbstractAlgebrasBirkhoff:SubdirectUnions
  • Definition 2.2
  • Theorem 2.3: Mal'tcev Malcev:AlgebraicSystems
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6: Clifford Clifford:PartiallyOrderedAbelianGroups
  • Theorem 2.7: Gurevich, Kokorin Gurevich-Kokorin:UniversalEquivalence
  • Theorem 2.8: Khisamiev Khisamiev:UniversalTheoryAbelianGroups
  • Theorem 2.9: Hölder's Theorem Holder:Axiome
  • Lemma 2.10
  • ...and 68 more