Remarks on the Valid Equations in Lattice Implication Algebras

We present a set of equations that axiomatizes the class of all lattice implication algebras. The construction is different from the one given in [7], and the proof is direct: i.e., it does not rely on results from outside the realm of the lattice implication algebras, such as the theory of BCK-algebras. Then we show that the lattice H implication algebras are nothing more than the familiar Boolean algebras. Finally we obtain some negative results for the embedding of lattice implication algebras into Boolean algebras.