Given (Z,*) is an algebraic structure. Hence it is closed.
Now, to become SEMI-GROUP: It should be associative
(x*y)*z = max(max(x,y),z) = max(x,y,z)
x*(y*z) = max(x,max(y,z)) = max(x,y,z)
eg. (1*2)*3 = max((max(1,2),3) = max(2,3) = 3
1*(2*3) = max(1,max(2,3)) = max(1,3) = 3
Hence (x*y)*z = x*(y*z), so ASSOCIATIVE i.e SEMI-GROUP
Now, to become MONOID: It should have an identity element
Let there be such an element and let the identity element be e.
So, as per definition: (a*e) = max(a,e) = a
Nice. But the question is: What if we take e=(a+1) ∀ a ∈ Z
Then, (a*(a+1)) = max(a,(a+1)) = (a+1) and NOT a
So, there can be no such identity(e) element possible.
Hence, its not a MONOID.
So correct choice is: (D) None