Subvarieties of pointed Abelian l-groups

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 $\Gamma$ 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.