Is the product of two elements in a monoid also an element of that monoid?

Question

Let $(M, ·)$ be a monoid and $x$ and $y$ any two elements of said monoid. Is the following true?

$$∀ x, y ∈ M, \qquad x · y ∈ M$$

Answer 1

A monoid is a triple $(M,\circ,e)$ consisting of a non-empty set $M$, an operation $\circ:M\times M\rightarrow M:(x,y)\mapsto x\circ y (=xy)$ and a neutral element $e\in M$. So the multiplication of two elements $x,y\in M$ yields another element $xy\in M$.