Semigroup/Examples/x+y+xy on Positive Integers

Example of Semigroup
Let $\circ: \Z_{\ge 0} \times \Z_{\ge 0}$ be the operation defined on the integers $\Z_{\ge 0}$ as:
 * $\forall x, y \in \Z_{\ge 0}: x \circ y := x + y + x y$

Then $\struct {\Z_{\ge 0}, \circ}$ is a semigroup.

Proof
We have that:
 * $\forall x, y \in \Z_{\ge 0}: x + y + x y \in \Z_{\ge 0}$

and so $\struct {\Z_{\ge 0}, \circ}$ is closed.

Now let $x, y, z \in \Z_{\ge 0}$.

We have:

and:

As can be seen by inspection:
 * $x \circ \paren {y \circ z} = \paren {x \circ y} \circ z$

and so $\circ$ is associative.

The result follows by definition of semigroup.