Superabelian logics
Petr Cintula, Filip Jankovec, Carles Noguera
TL;DR
The paper develops a unified algebraic framework for superabelian logics based on Abelian $\ell$-groups and clarifies the relationships among $\mathrm{Ab}$, $\mathrm{Lu}$, and $\mathrm{Lu}_{\infty}$ while introducing two major expansion families: infinitary extensions of $\mathrm{Ab}$ and pointed Abelian logic $\mathrm{pAb}$ with a new constant. It proves that $\mathrm{Ab}$ has no nontrivial finitary extensions, while a rich lattice of infinitary extensions exists, including a full axiomatization of the real-number extension $\vDash_{\boldsymbol{\mathit{R}}}$ and a hierarchy $\mathrm{Ab} \subsetneq \vDash_{\boldsymbol{\mathit{R}}} \subsetneq \vDash_{\boldsymbol{\mathit{Q}}} \subsetneq \vDash_{\boldsymbol{\mathit{Z}}}$, with $2^{2^\omega}$ distinct logics between $\vDash_{\boldsymbol{\mathit{R}}}$ and $\vDash_{\boldsymbol{\mathit{Q}}}$. The pointed expansion $\mathrm{pAb}$ yields a robust, finitely strongly complete family with canonical pointed triples $\{\mathrm{R}_{-1},\mathrm{R}_{0},\mathrm{R}_{1}\}$ (and analogous $\mathrm{Q}$-triples), while Lukasiewicz unbound logic $\mathrm{Lu}$ and its infinitary variant $\mathrm{Lu}_{\infty}$ provide concrete translations to (and from) standard Łukasiewicz logic via $\tau$. Overall, the work supplies a unified framework for axiomatizing logics of prominent pointed abelian groups and sets foundations for extending these methods to other structured logics.
Abstract
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 \emph {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 $2^{2^ω}$ distinct logics in this family. Second, we introduce \emph{pointed Abelian logic} (pAb), the logic of pointed Abelian lattice-ordered groups, by adding a new constant to the language. This framework includes \emph{Łukasiewicz unbound logic}. We provide axiomatizations for its finitary and infinitary versions as extensions of pAb and establish their precise relationship with standard Łukasiewicz logic via a formal translation. Finally, the methods developed for this analysis are generalized to axiomatize the logics of other prominent pointed groups.