Algebraic Expansions of Logics

Abstract

An algebraically expandable (AE) class is a class of algebraic structures axiomatizable by sentences of the form ∀∃!⋀p=q . For a logic L algebraized by a quasivariety Q we show that the AE-subclasses of Q correspond to certain natural expansions of L, which we call 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 ℓ -groups and perfect MV-algebras, thus fully describing the algebraic expansions of their associated logics.