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 $\paren {y - 1} \le 0$.
\(\ds \paren {x - 1}\) | \(\le\) | \(\ds 0\) | ||||||||||||
\(\ds \paren {y - 1}\) | \(\le\) | \(\ds 0\) |
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.
$\blacksquare$
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $5$: Semigroups: Exercise $1$