* 1 2 3 4
1 1 1 1 1
2 1 2 3 4
3 1 3 3 2
4 2 4 3 4
So i found the identity element which is 2. But i'm wondering if there is any invertible elements?
The identity element is always invertible, marked in green. Check the elements marked in red; there is no element other than 2 that is invertible, only left-invertible or right-invertible.
Edit: I should clarify: even though the element 4 has both a right and left inverse, since it does not have a two sided inverse, it is not invertible. This actually proves that associativity does not hold; notice that (3 * 4) * 1 does not equal 3 * (4 * 1).