Subsemigroup/Examples/x+y-xy on Integers

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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 $\struct {\Z, \circ}$.


Proof

It is established in Operation Defined as $x + y - x y$ on Integers that $\struct {\Z, \circ}$ is a semigroup.

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 $\paren {y - 1} \le 0$.
\(\displaystyle \paren {x - 1}\) \(\le\) \(\displaystyle 0\)
\(\displaystyle \paren {y - 1}\) \(\le\) \(\displaystyle 0\)


So:

\(\displaystyle \paren {x - 1} \paren {y - 1}\) \(\ge\) \(\displaystyle 0\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle x y - x - y + 1\) \(\ge\) \(\displaystyle 0\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle x y - x - y\) \(\ge\) \(\displaystyle -1\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle x + y - x y\) \(\le\) \(\displaystyle 1\)

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.

$\blacksquare$


Sources