Superabelian logics

This paper presents a unified algebraic study of a family of logics related to Abelian logic (Ab), the logic of Abelian lattice-ordered groups. We treat Ab as the base system and refer to its expansions as superabelian logics. The paper focuses on two main families of expansions. First, we investigate the rich landscape of infinitary extensions of Ab, providing an axiomatization for the infinitary logic of real numbers and showing that there exist $2^{2^\omega}$ distinct logics in this family. Second, we introduce pointed Abelian logic (pAb), the logic of pointed Abelian lattice-ordered groups, by adding a new constant to the language. This framework includes {\L}ukasiewicz unbound logic. We provide axiomatizations for its finitary and infinitary versions as extensions of pAb and establish their precise relationship with standard {\L}ukasiewicz logic via a formal translation. Finally, the methods developed for this analysis are generalized to axiomatize the logics of other prominent pointed groups.