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.

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:

$\ds x \circ \paren {y \circ z}$

$=$

$\ds x + \paren {y \circ z} + x \paren {y \circ z}$

Definition of $\circ$

$\ds $

$=$

$\ds x + \paren {y + z + y z} + x \paren {y + z + y z}$

Definition of $\circ$

$\ds $

$=$

$\ds x + y + z + y z + x y + x z + x y z$

and:

$\ds \paren {x \circ y} \circ z$

$=$

$\ds \paren {x \circ y} + z + \paren {x \circ y} z$

Definition of $\circ$

$\ds $

$=$

$\ds \paren {x + y + x y} + z + \paren {x + y + x y} z$

Definition of $\circ$

$\ds $

$=$

$\ds x + y + x y + z + x z + y z + x y z$

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.

$\blacksquare$

1951: Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts ... (previous) ... (next): Chapter $\text{I}$: Semi-Groups and Groups: $1$: Definition and examples of semigroups: Example $3$