Suppose (M,*) is a monoid with an element q such that q*q=q. Can we conclude that q is the identity element for M? If so, provide a proof. If not, provide a counterexample. Solution yes if q*q =q, then q is the identity element. as we know that monoid holds the associative law,and also has a identity element e, then e*q=q*e=q now it is given that q*q=q, hence q*q=q*e then by left cancellation law q=e.