Does semigroup and monoid have to be closed under the binary operation?

Question

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!

Answer 1

Semigroup and monoid have to be closed under the binary operation. You can conclude it from wiki's definition of a binary operation as "a mapping of the elements of the Cartesian product $S \times S$ to $S$" and from wiki's definition of associativity as $$(x ∗ y) ∗ z = x ∗ (y ∗ z),\ \forall x, y, z \in S$$ But it is better to state it explicitly that the set S is closed under the operation $*$. Because Wiki defines "an n-ary operation $\ \omega$ from $X_1\ldots, X_n$ to $Y$ [as] a function $\omega: X_1 \times \ldots \times X_n \to Y$". From this one would conclude that a binary operation from $X_1\, X_2$ to $Y$ must not be closed.

Answer 2

Closure is needed for all of them, but it's often called totality. See this table, which you'll find here & here.