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\cdot5=\boxed{20} $.