The given binary operation is a semi-group but not a monoid, as it doesn't have a unique identity element.
Let us assume e, be an identity element for the given binary operation. Then the following must hold true for all Xϵℤ:
X*e = e*X = X ;
Now, we know that X*Y= X and obviously it isn't commutative, which means e must be X itself. (X*X=X)
But, this means that every distinct element of the set ℤ has itself as its identity element.
Therefore, no identity element exists for the given binary operation.
