 An interesting result on the finitary logics which are fully selfextensional logics with a conjunction or with the deduction-detachment property for a single term is that their class of algebras \(\bAlg\bL\) is always a variety. It looks surprising that many finitary and finitely algebraizable logics have a variety as its equivalent algebraic semantics, when the theory of algebraizable logics allows in general to prove only that the equivalent algebraic semantics of a finitary and finitely algebraizable logic is a quasivariety. The result explains this phenomenon for the finitary and finitely algebraizable logics to which it applies. For many other finitary and finitely algebraizable logics to find a convincing explanation is still an open area of research.