Exercise
Is the union of two submonoids always a submonoid?
Yes
Sometimes
Never
No. Think of Z2 with ordinary addition and take
the submonoids {(a,0) | a an integer} and {(0,a) | a an integer}. Then (1,0) belongs to the first submonoid, (0,1) to the
second, but (1,1) does not belong to the union of the two. So the
union is not even closed with respect to the operation.
Indeed, it can happen that one takes two submonoids that are the same.
No, think of trivial exceptions