Algebraic expansions of logics and algebras and a case study of Abelian l-groups and perfect MV-algebras

TL;DR

This paper explores algebraic expansions of logics through AE-classes, fully characterizing their structure in Abelian -groups and perfect MV-algebras, linking algebraic and logical frameworks.

Contribution

It introduces the concept of algebraic expansions for logics, characterizes AE-subclasses of specific algebraic structures, and connects these to natural logical expansions.

Findings

AE-classes correspond to algebraic expansions of logics

Complete characterization of AE-subclasses for Abelian -groups

Complete characterization of AE-subclasses for perfect MV-algebras

Abstract

An algebraically expandable (AE) class is a class of algebraic structures axiomatizable by sentences of the form $\forall \exists! \land p = q$. For a logic $L$ algebraized by a quasivariety $\mathcal{Q}$ we show that the AE-subclasses of $\mathcal{Q}$ correspond to certain natural expansions of $L$, which we call {\em algebraic expansions}. These turn out to be a special case of the expansions by implicit connectives studied by X. Caicedo. We proceed to characterize all the AE-subclasses of Abelian $\ell$-groups and perfect MV-algebras, thus fully describing the algebraic expansions of their associated logics.