Proof for P-complete is not closed under intersection
Let $A$ be any P-complete problem (say circuit evaluation). Here are two other P-complete problems: $A_0 = \{0x : x \in A \}$ and $A_1 = \{1x : x \in A\}$. The problem $A_0 \cap A_1 = \emptyset$, while definitely in P, isn't P-complete. The latter is true even if we allow Turing reductions, assuming $L \neq P$.