Subvarieties of pointed Abelian l-groups

Abstract

This paper provides a complete classification of all subvarieties of pointed Abelian lattice-ordered groups (l-groups), as well as all subquasivarieties that are generated by their totally ordered members. We present two complementary approaches to achieve this classification. First, using purely l-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 l-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 l-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 an extended version of 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 l-groups. As a direct consequence, we obtain a complete structural description of the lattice of subquasivarieties that are generated by their totally ordered members.