Subvarieties of pointed Abelian l-groups

This paper provides a complete classification of the subvarieties and subquasivarieties of pointed Abelian lattice-ordered groups (-groups) that are generated by chains. We present two complementary approaches to achieve this classification. First, using purely -group-theoretic methods, we analyze the structure of lexicographic products and radicals to identify all join-irreducible members of the lattice of subvarieties of positively pointed -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 celebrated 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 MV-chains, 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. These results offer a comprehensive structural understanding of one of the most fundamental classes of ordered algebraic structures.