 For an arbitrary logic \(\bL\), the relation between the classes \(\bAlg\bL\) and \(\bAlg^*\bL\) is that \(\bAlg\bL\) is the closure of \(\bAlg^*\bL\) under subdirect products, in particular \(\bAlg^*\bL \subseteq \bAlg\bL\). In general, the two classes may be different. For example, if \(\bL\) is the \((\wedge , \vee)\)-fragment of classical propositional logic, \(\bAlg\bL\) is the variety of distributive lattices (the class that has been always taken to be the natural class of algebras associated with \(\bL)\) while \(\bAlg^*\bL\) is properly included in it —in fact \(\bAlg^*\bL\) is not a quasivariety. Nonetheless, for many logics \(\bL\), in particular for the algebraizable and the protoalgebraic ones to be discussed in the next sections, and also when \(\bAlg^*\bL\) is a variety, the classes \(\bAlg\bL\) and \(\bAlg^*\bL\) are equal. This fact can explain why in the 1980s, before the algebraic study of non-protoalgebraic logics was considered worth to be pursued, the conceptual difference between the two definitions was not needed and, accordingly, it was not considered (or even discovered).