Superabelian logics

Abstract: This paper presents an algebraic study of a family of logics related to Abelian logic ($Ab$), the logic of Abelian lattice-ordeded groups. $Ab$ is treated as the basic, weakest logic, and its expansions are referred to as superabelian logics. To account for the language differences in Lukasiewicz logics, we introduced the expansion of $Ab$ with an additional constant and call it pointed Abelian logic ($pAb$), with semantics based on pointed Abelian $l$-groups. We take the first steps in the study of infinitary extensions of $Ab$, give a completeness theorem for $pAb$, and obtain axiomatizations of Lukasiewicz unbound logics relative to $pAb$. The paper concludes with an exploration of other significant extensions of $pAb$ and their algebraic characterizations.