As stated in the title, I am wondering whether semigroup and monoid have to be closed under the binary operation. The reason I am asking about this is that in wiki pages of semigroup and monoid, the property of CLOSED is not mentioned in the definition, while CLOSED is an essential property of group, starting at the beginning of the definition of group.
Is wiki wrong about it?
Much appreciation for any help!
Best Answer
Closure is needed for all of them, but it's often called totality. See this table, which you'll find here & here.