The maximal cardinality of the base in $P_{2}\times P_{2}$

Some structural properties are discussed of the functions in $P_{2}\times P_{2}$. We describe the properties of functions not belonging to the maximal subalgebras $R_{4}$, $ R_{5}$, and $R_{11}$ and show that the maximal cardinality of the basis in $P_{2}\times P_{2}$ is equal to $8$.