Semigroup/Examples/x+y-xy on Integers

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Example of Semigroup

Let $\circ: \Z \times \Z$ be the operation defined on the integers $\Z$ as:

$\forall x, y \in \Z: x \circ y := x + y - x y$

Then $\struct {\Z, \circ}$ is a semigroup.


We have that:

$\forall x, y \in \Z: x \circ y \in \Z$

and so $\struct {\Z, \circ}$ is closed.

Now let $x, y, z \in \Z$.

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 $\circ$
\(\ds \) \(=\) \(\ds x + y + z - y z - x y - x z + x y z\)


\(\ds \paren {x \circ y} \circ z\) \(=\) \(\ds \paren {x \circ y} + z - \paren {x \circ y} z\) 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.


Also see