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

An algebraically expandable (AE) class is a class of algebraic structures axiomatizable by sentences of the form $\mathrm{\forall }\mathrm{\exists }!\wedge p=q$. 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 $\mathrm{\ell }$-groups and perfect MV-algebras, thus fully describing the algebraic expansions of their associated logics.