Subsemigroup/Examples/x+y-xy on Integers

Example of Subsemigroup

Let $\struct {\Z, \circ}$ be the semigroup where $\circ: \Z \times \Z$ is the operation defined on the integers $\Z$ as:

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

Let $T$ be the set $\set {x \in \Z: x \le 1}$.

Then $\struct {T, \circ}$ is a subsemigroup of $\struct {\Z, \circ}$.

From the Subsemigroup Closure Test it is sufficient to demonstrate that $\struct {T, \circ}$ is closed.

Let $x, y \in \Z_{\le 1}$.

$\paren {x - 1}\le 0$ and $\paren {y - 1} \le 0$.

$\ds \paren {x - 1}$

$\le$

$\ds 0$

$\ds \paren {y - 1}$

$\le$

$\ds 0$

So:

$\ds \paren {x - 1} \paren {y - 1}$

$\ge$

$\ds 0$

$\ds \leadsto \ \ $

$\ds x y - x - y + 1$

$\ge$

$\ds 0$

$\ds \leadsto \ \ $

$\ds x y - x - y$

$\ge$

$\ds -1$

$\ds \leadsto \ \ $

$\ds x + y - x y$

$\le$

$\ds 1$

So:

$x \le 1, y \le 1 \implies x \circ y \le 1$

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

The result follows by definition of subsemigroup.

$\blacksquare$