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](https://en.wikipedia.org/wiki/Semigroup) and [monoid](https://en.wikipedia.org/wiki/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](https://www-users.cse.umn.edu/%7Ebrubaker/docs/152/152groups.pdf).

Is wiki wrong about it?

Much appreciation for any help!