In [3] Balbes and Grw gave intrinsic and extrinsic characterizations of injective Stone algebras. Specifically, the injectives can be characterized (extrinsically) as those Stone algebras that are the direct product of a complete Boolean algebra with a complete Post algebra of order three. In this paper we give an characterization of those lattices that are the direct product of a Boolean algebra with a Post algebra of order n; calling such a lattice a Post-like algebra of order n. Specializing, we give a new equational characterization of Post algebras of order n. In addition, Mycielski's question 'Is every equationally compact algebra the retract of a compact topological algebra?' (see [13]) is answered affirmatively for the class ~ of all Post-like algebras of order n. In fact, the equationally compact algebras in ~7. are shown to be the complete algebras. Characterizations of completely distributive Post-like algebras are also given. Restricting attention to Post-like algebras of order three, we give a nice (intrinsic) characterization of injective Stone algebras.