jmc

The first possibility we have is that $A$ and $B$ are both equal to 0, in which case the addition problem would simplify to $0+0=0$. However, since the problem states that both $A$ and $B$ are positive integers, we can eliminate this possibility. Therefore we can assume that there is carrying in the right column, giving us $A_9+B_9=10_9$ or that $A+B=9$. Since we know that there is a 1 carried over, the left column tell us that $1+A=B$. Solving this system of equations, we find that $A=4$ and $B=5$. Thus, $A\cdot B=4\cdot 5=\boxed{20}$.