We introduce the notion of an everywhere strongly logifiable algebra: a finite non-trivial algebra A such that for every F 2 P(A) r f;;Ag the logic determined by the matrix hA; Fi is a strongly algebraizable logic with equivalent algebraic semantics the variety generated by A. Then we show that everywhere strongly logifiable algebras belong to the field of universal algebra as well as to the one of logic by characterizing them as the finite non-trivial simple algebras that are constantive and generate a congruence distributive and n-permutable variety for some n > 2. This result sets everywhere strongly logifiable algebras surprisingly close to primal algebras. Nevertheless we shall provide examples everywhere strongly logifiable algebras that are not primal. Finally, some conclusion on the problem of determining whether the equivalent algebraic semantics of an algebraizable logic is a variety is obtained.
