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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2509.05044 |
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| _version_ | 1866917367312809984 |
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| author | Jankovec, Filip |
| author_facet | Jankovec, Filip |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_05044 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Subvarieties of pointed Abelian l-groups Jankovec, Filip Logic Logic in Computer Science 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. |
| title | Subvarieties of pointed Abelian l-groups |
| topic | Logic Logic in Computer Science |
| url | https://arxiv.org/abs/2509.05044 |